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Strict chordal and strict split digraphs
Discrete Applied Mathematics (
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Strict chordal and strict split digraphs✩ Pavol Hell a,∗ , César Hernández-Cruz a,b a
School of Computing Science, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6
b
Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad Universitaria, C.P. 04510, México, D.F., Mexico
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Article history: Received 14 April 2015 Received in revised form 9 February 2016 Accepted 11 February 2016 Available online xxxx
abstract We introduce new versions of chordal and split digraphs, and explore their similarity with the corresponding undirected notions. © 2016 Elsevier B.V. All rights reserved.
Keywords: Split digraph Chordal digraph Perfect digraph Recognition algorithm
1. Introduction A graph G is chordal if its vertices can be linearly ordered by < so that for any u < v < w , if u ∼ v and u ∼ w then v ∼ w . Such an ordering < is called a perfect elimination ordering of G. Chordal graphs are an important class of perfect graphs, admit elegant recognition algorithms and characterizations, efficient optimization algorithms, and interesting applications [17]. They arise, for example, from consideration of sparse symmetric matrices whose sparseness can be preserved during Gaussian elimination [34]. A graph is chordal if and only if it does not have an induced cycle of length greater than three [17]. Chordal graphs can be recognized in linear time [34]; in fact the algorithm can be made certifying, in the sense that it either finds a simplicial ordering or an induced cycle of length greater than three [35]. The chromatic number, as well as the size of a maximum clique and maximum independent set in a chordal graph can be found in linear time, from the perfect elimination ordering [15,17]. Even nicer properties hold for its subclass of split graphs. A graph is split if its vertex set can be partitioned into a clique and an independent set [13]. A graph G is split if and only if both G and G are chordal [17]. It can be seen from this that a graph is split if and only if it does not have an induced C4 , C5 , or C4 [13]. Further, a graph is split if and only if its vertices can be linearly ordered by < so that for any u < v < w , if u ∼ v then v ∼ w [10]. Whether or not an input graph is split can be recognized in linear time just by considering its degree sequence [17,13]. A certifying linear time recognition algorithm for split graphs is given in [24,23]. It is known that the proportion of chordal graphs on n vertices that are split graphs tends to one as n increases [2]. Perhaps the best known subclass is the class of interval graphs, since it has the most numerous natural applications [3,8]. A graph G is an interval graph if its vertices v can be represented by intervals Iv so that v ∼ w if and only if Iv intersects Iw [17]. A graph is an interval graph if and only if its vertices can be linearly ordered by < so that for any u < v < w , if u ∼ w
✩ This research was supported by a research grant from NSERC Canada and ERCCZ LL 1201.
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Corresponding author. E-mail addresses: [email protected] (P. Hell), [email protected] (C. Hernández-Cruz).
http://dx.doi.org/10.1016/j.dam.2016.02.009 0166-218X/© 2016 Elsevier B.V. All rights reserved.
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then v ∼ w [17]. (It follows that an interval graph is necessarily chordal.) A forbidden induced subgraph characterization appears in [28] (see also [16,14]), and there are several linear-time recognition algorithms [5,25,19,9], and certifying algorithms [26]. All these graph classes are contained in the class of perfect graphs. A graph G is perfect if the chromatic number and the maximum clique size are equal for G and all of its induced subgraphs. The class of perfect graphs is considered of prime importance in algorithmic graph theory [4,17,7], a prototype framework for combinatorial max–min results. A graph is perfect if and only if it does not contain an induced cycle of odd length greater than three, or its complement [4,7]. Perfect graphs can be recognized in polynomial time [6], and the basic optimization problems of finding the chromatic number, the clique number, and the independence number can be solved in polynomial time [18]. There have been attempts to translate the elegance of these results to the realm of digraphs. Perhaps the best known is the digraph analogue of interval graphs, defined as follows [11]. (A different definition was proposed in [20]; it only yields acyclic digraphs.) A digraph D is an interval digraph if its vertices v can be represented by pairs of intervals Iv , Jv , so that v → w if and only if Iv intersects Jw . Some characterizations of interval digraphs are known [11], especially in terms of their adjacency matrices; however there is no forbidden induced subgraph (or substructure) characterization known, and the existing polynomial time recognition algorithms have high-degree-polynomial time bounds [31]. (We note that a low-degree-polynomial time algorithm is claimed in [33].) Interestingly, there is a more convenient subclass of interval digraphs. In [12] the authors define a digraph D to be an adjusted interval digraph if its vertices v can be represented by pairs of intervals Iv , Jv , so that v → w if and only if Iv intersects Jw , where the intervals Iv and Jv have the same left endpoint (are ‘‘left-adjusted’’). It turns out that adjusted interval digraphs show a greater similarity to interval graphs than plain interval digraphs. In particular, they have a forbidden structure characterization, an ordering characterization, and a low-degree-polynomial time recognition algorithm [12]. It turned out that interval digraphs were too ambitious a generalization of interval graphs, and by being more modest (while still making a large digraph generalization of interval graphs), we obtain a better analogue. Chordal digraphs are defined in [30,22], also in a way to correspond to sparse general matrices (not necessarily symmetric as above), whose sparseness can be preserved during Gaussian elimination. (Again a different definition tailored to acyclic digraphs was proposed in [21].) Specifically, a digraph D is a chordal digraph if its vertices can be linearly ordered by < so that for any u < v < w , if v → u and u → w then v → w [30,22]. Some natural analogues of the results about chordal graphs are obtained for the class of these chordal digraphs [30]. However, a forbidden subgraph characterization is not known, except in certain very special cases [30,21,29]. Split digraphs were defined in [27] as follows. D is a split digraph if its vertices can be partitioned into four sets A, B, C , D where A is a strong clique (a complete symmetric digraph), D is an independent set, all possible arcs go from A to C and from B to A and C , and no arcs go from C to B or D and from D to B. Moreover, it is required that the partition does not place all vertices in B or all vertices in C . (Otherwise every digraph would be split.) The main appeal of these split digraphs seems to be that can be recognized by just considering their degree sequence, just as split graphs can [27]. However, this class of split digraphs is not closed under taking induced subgraphs; any digraph D can be made split by the addition of one new vertex dominating all vertices of D. It appears that these digraph analogues of chordal and split graphs may again be too general to recover the elegance of chordal and split graphs. In this paper, we propose two new notions of chordal and split digraphs, which we call strict chordal digraphs and strict split digraphs. These properly generalize the undirected cases, in the sense that a chordal (respectively split) graph, when viewed as a digraph, by replacing each edge uv by the two arcs uv and v u, is a strict chordal (respectively strict split) digraph. There is however a much greater number of strict chordal and strict split digraphs that do not arise this way from their undirected analogues (see the Concluding Remarks). Nevertheless, the underlying graph of any strict chordal digraph is a chordal graph, and the underlying graph of any strict split digraph is a split graph. Moreover our classes turn out to be close enough to the undirected notions to allow extending some of the fundamental theorems from the undirected case, including forbidden subgraph characterizations and polynomial time recognition algorithms (linear time in the case of strict split graphs). The class of perfect graphs also has a digraph analogue. For digraphs, D is a perfect digraph [1] if for D and every induced subgraph of D, the dichromatic number and the clique number are equal. (The dichromatic number of D is the smallest number of colours for which D can be coloured so that each colour induces an acyclic subgraph; and the clique number of D is the number of vertices in a largest strong clique.) The authors of [1] have succeeded in obtaining many results analogous to the undirected cases, including a forbidden subgraph characterization, and polynomial time optimization algorithms for some basic problems. However, the recognition of perfect digraphs is N P -complete [1]. It will turn out that strict chordal digraphs and (therefore) the strict split digraphs are perfect in this sense. As used above, in a graph G we write u ∼ v to mean there is an edge joining u and v in G; we also say that u and v are adjacent. Similarly, in a digraph D, u → v means there is an arc from u to v ; we also say that u dominates v . In addition to these standard terms we also use the following helpful terminology, in a digraph D. We say that u and v are adjacent in (a digraph) D if u → v or v → u (or both); in analogy with graphs we write u ∼ v in this case. We say that u and v are strongly adjacent in D if u → v and v → u; in this case we write u ↔ v . (We also call u ↔ v a digon or a symmetric arc.) We say that u and v are weakly adjacent in D if either u → v or v → u; in this case we write u ÷ v . (In the special case when u → v but not v → u we write u → v and similarly for v → u.)
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In the figures adjacency u ∼ v is depicted by a solid line joining u and v , strong adjacency u ↔ v by a solid line joining u and v and having arrows at both ends, and weak adjacency u ÷ v by a (slightly thinner) solid line joining u and v and marked by a short bar. A strong clique in a digraph D is a set of vertices in which any two distinct vertices are strongly adjacent; a tournament is a set of vertices in which any two distinct vertices are weakly adjacent; and a semi-complete digraph is a set of vertices in which any two distinct vertices are adjacent. Let D be a digraph and G the undirected graph on the same vertices with u ∼ v in G if and only if u ∼ v in D. We call G the underlying graph of D, and also call D a super-orientation of G. An orientation of G is a super-orientation without strongly adjacent vertices (digons). A bi-orientation of G is a super-orientation without weakly adjacent vertices (all arcs are digons). 2. Strict chordal digraphs Recall that a digraph is chordal [22,30] if its vertices can be linearly ordered by < so that for any u < v < w , if v → u and u → w then v → w . We propose the following variant of this notion. A digraph is strict chordal if its vertices can be linearly ordered by < so that for any u < v < w, if v ∼ u and u ∼ w then v ↔ w . In other words, if < orders the vertices as v1 , v2 , . . . , vn , then the set of vertices from vi+1 , vi+2 , . . . , vn that are adjacent with vi form a strong clique. In particular, v1 is adjacent with a set of vertices that form a strong clique; such a vertex is called simplicial. Note that strict chordal digraphs are a restricted class of chordal digraphs. Note also that, as suggested earlier, the biorientation of any chordal graph is a strict chordal digraph. In fact, it turns out that strict chordal digraphs are a non-trivial and interesting class of digraphs, properly more general than just bi-orientations of undirected chordal graphs, but still well enough behaved. In particular, we shall provide a forbidden induced structure characterization for them. First we note that there is a polynomial time recognition algorithm for strict chordal digraphs. Indeed, since an induced subgraph of a strict chordal digraph is again strict chordal, we can test whether or not D is a strict chordal digraph by finding a simplicial vertex v of D and testing recursively that D − v is strict chordal. A strict chordal obstruction is a digraph which is not a strict chordal digraph. A minimal strict chordal obstruction is a strict chordal obstruction D such that every proper induced subgraph of D is a strict chordal digraph. Observe that the underlying graph of a strict chordal digraph D must be chordal. Moreover, the graph formed by the symmetric edges (digons) of a strict chordal digraph D must be a chordal graph. In fact, the ordering < from the definition of a strict chordal digraph D is a perfect elimination ordering of both the underlying graph and the graph formed by the symmetric edges of D. We now list some basic minimal strict chordal obstructions. It follows from the last remarks that any super-orientation of a cycle of length greater than three is a minimal strict chordal obstruction. Moreover, any super-orientation of the four-cycle C4 with a chord that is not a digon is a strict chordal obstruction. Indeed, the first vertex in any ordering < cannot be incident to the chord, since the same ordering must be a perfect elimination ordering for the underlying graph; thus the first vertex is one of the other two vertices, and consequently the chord should have been a digon. It is easy to see that it is in fact a minimal strict chordal obstruction. Therefore we denote by C the family of minimal strict chordal obstructions consisting of all super-orientations of cycles of length greater than three, together with all super-orientations of C4 with a chord that is not a digon. (See the upper part of Fig. 1; the conventions concerning the various kinds of lines depicting the arcs are detailed at the end of the previous section.) It is easy to see that the definition of < implies that any tournament on at least three vertices is a strict chordal obstruction. (In an ordering < the second and third vertex need to be strongly adjacent.) Therefore the two three-vertex tournaments (the directed three-cycle and the transitive triple) are minimal strict chordal obstructions. Similarly, any semi-complete digraph with n vertices that is a strict chordal digraph must contain a strong clique containing n − 1 vertices (excluding the first vertex in the ordering <). Thus any four-vertex semi-complete digraph without a strong clique on three vertices is a strict chordal obstruction. Let D denote the family consisting of both tournaments on three vertices and all four-vertex semi-complete digraphs without a strong clique on three vertices. (The lower part of Fig. 1 depicts all minimal members of D , up to symmetry. Note that several four-vertex digraphs are not minimal because they contain a tournament on three vertices.) There are two other infinite families of forbidden induced subgraphs. The lollipop on vertices a0 , a1 , a2 , a3 , . . . , ak , k ≥ 2, has arcs a0 ↔ a1 , a0 ÷ a2 , a1 ÷ a2 , and a2 ∼ a3 ∼ a4 · · · ∼ ak . The vertex ak is called the end-vertex of the lollipop. The sum of a lollipop on a0 , a1 , a2 , a3 , . . . , ak and a lollipop on b0 , b1 , b2 , b3 , . . . , bℓ is obtained from their disjoint union by identifying ak = bℓ . The join of these two lollipops is obtained from their disjoint union by the addition of all digons ak ↔ bi , i = 0, 1, 2, . . . , ℓ and bℓ ↔ aj , j = 0, 1, 2, . . . , k. The family L consists of all sums of lollipops, and the family K consists of all joins of lollipops, see Fig. 2. It turns out that families C , D , L, and K contain all the minimal obstructions to strict chordal digraphs. We proceed to prove our forbidden induced subgraph characterization of strict chordal digraphs. Theorem 1. D is a strict chordal digraph if and only if it does not contain an induced subgraph from C ∪ D ∪ L ∪ K . Proof. We have already shown all digraphs in C ∪ D ∪ L ∪ K are strict chordal obstructions. Therefore, let D be a digraph without induced subgraphs from the families C , D , L, and K ; we shall prove that D is strict chordal. Note that the underlying graph of D is chordal, since D contains no induced subgraph from C .
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Fig. 1. The minimal strict chordal obstruction families C and D .
Fig. 2. The minimal obstruction families L and K built of lollipops.
Lemma 2. If D does not contain an induced subgraph from C ∪ D ∪ L ∪ K and is not semi-complete, then any minimal cutset of D is a strong clique. Proof. Let S be a minimal cutset of D. Since the underlying graph of D is chordal, it follows from a corresponding result on chordal graphs [17] that S is a semi-complete digraph. If S is not a strong clique, then it contains weakly adjacent vertices u and v . Since S is a minimal cutset of D, there are components D1 and D2 of D − S such that u and v are adjacent with vertices in both D1 and D2 . Let Pi be a shortest uv -path all whose internal vertices lie in Di , i ∈ {1, 2}. Then either Pi ∪ {u, v} induces a chordless cycle for some i ∈ {1, 2}, or P1 ∪ P2 ∪ (u, v) induces a four-cycle with a chord that is not a digon; both are in the family C , a contradiction. Note that a strict chordal digraph always contains a simplicial vertex (the first vertex in the ordering <). Lemma 3. If a strict chordal digraph D does not contain two non-adjacent simplicial vertices, then D is semi-complete, or every simplicial vertex of D is the end-vertex of an induced lollipop.
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Proof. We will proceed by induction on the number of vertices of D. The result is easy to verify for strict chordal digraphs D with fewer than five vertices. Let D be a non-semi-complete connected chordal digraph on n ≥ 5 vertices. Let x be a simplicial vertex of D and let y be a vertex non-adjacent with x. By Lemma 2, a minimal cutset S separating x from y must be a strong clique. Let X and Y be the components of D − S containing x and y, respectively. Every simplicial vertex of D[Y ∪ S ] belongs to S, otherwise D would have two non-adjacent simplicial vertices. Let u be a simplicial vertex of D[Y ∪ S ]. Suppose first that D[Y ∪ S ] is semi-complete. Since S is a strong clique, there are u1 , u2 ∈ Y \ S such that u ÷ u1 and u ÷ u2 (otherwise some vertex in Y would be a simplicial vertex of D). Now, if P is an induced xu-path in the underlying graph of D, then V (P ) ∪ {u1 , u2 } induces a lollipop in D with x as its end-vertex. If D[Y ∪ S ] is not semi-complete, then, by the induction hypothesis, it contains an induced lollipop with end vertex u ∈ S. Again, if P is an induced xu-path in the underlying graph of D, and Z are the vertices of the induced lollipop of D[Y ∪ S ], then V (P ) ∪ Z induces a lollipop in D with end-vertex x. We note that for semi-complete digraphs D, if there is a unique arc that is not a digon then the two vertices on the arc are both simplicial. Otherwise, if there are at least two arcs that are not digons, there is at most one simplicial vertex. We are now ready to show, by induction on the number of vertices of D, that every C ∪ D ∪ L ∪ K -free digraph has a simplicial vertex. The result is again easy to verify for fewer than five vertices. Let D be a C ∪ D ∪ L ∪ K -free digraph with n vertices, n ≥ 5. If D is semicomplete, since it is D -free, it must contain a strong clique on n − 1 vertices as an induced subgraph. Observe that the remaining vertex is a simplicial vertex of D. Otherwise, let x and y be non-adjacent vertices in D, and let S be a minimal vertex cut separating x from y. If X and Y are the components of D − S containing x and y, respectively, then D1 = D[X ∪ S ] and D2 = D[Y ∪ S ] are strict chordal digraphs. If D1 or D2 contains two non-adjacent simplicial vertices, then D contains a simplicial vertex. So, suppose that every simplicial vertex of each D1 and D2 belongs to S. We claim that this cannot happen. Let s1 and s2 be simplicial vertices in D1 and D2 , respectively. There are three cases. In the first case, both D1 and D2 are semi-complete digraphs. Let u1 , u2 and v1 , v2 be the vertices weakly adjacent with s1 and s2 , respectively. Hence, {s1 , s2 , u1 , u2 , v1 , v2 } induces a digraph from L if s1 = s2 , or a digraph from K otherwise. In the second case, assume that D1 is semi-complete and D2 is not. If s1 = s2 , then, D contains an induced subgraph isomorphic to an element of L. Otherwise, let H be the lollipop on a0 , . . . , ak = s2 induced in D2 . Suppose that s1 ∼ a0 . Since D is C -free, s1 ↔ ai for every 0 ≤ i ≤ k. Let u1 , u2 ∈ X be such that s1 ÷ u1 and s1 ÷ u2 . Clearly, V (H ) ∪ {s1 , u1 , u2 } induces in D a digraph from K . Therefore, s1 is non-adjacent with a0 and a1 . But this implies that D contains an element of L as an induced subgraph. In the third case, neither D1 nor D2 is semi-complete. Let H1 be the lollipop on a0 , . . . , ak = s1 induced in D1 , and let H2 be the lollipop on b0 , . . . , bl = s2 induced in D2 . If s1 = s2 , then D[H1 ∪ H2 ] belongs to L. Else, if s1 is not adjacent with b0 or s2 is not adjacent with a0 , then H1 ∪ H2 contains an induced copy of a digraph in L. Assume that s1 is adjacent with b0 and s2 is adjacent with a0 . Since D is C -free, s1 ↔ bi for 0 ≤ i ≤ l, and s2 ↔ ai for 0 ≤ i ≤ k. Hence, D[H1 ∪ H2 ] belongs to K . Therefore at least one of D1 , D2 must have a simplicial vertex not contained in S. Hence, D has a simplicial vertex. By repeating the argument recursively we obtain the desired ordering <, proving that D is a strict chordal digraph. 3. Strict split digraphs A digraph D is a strict split graph if its vertices can be partitioned into a strong clique and an independent set. Such a partition will be called a strict split partition of D. Although the class of strict split digraphs is smaller than the class of split digraphs introduced in [27], it contains all biorientations of split graphs, and has interesting properties comparable to the class of split graphs. In particular, it admits a characterization by a finite set of forbidden induced subgraphs, an ordering characterization, and is the intersection of the class of strict chordal digraphs and the class of their complements. It is also the case that the underlying graph of a strict split digraph is a split graph. A strict split obstruction is a digraph which is not a strict split digraph. A minimal strict split obstruction is a strict split obstruction such that every proper induced subgraph is a strict split digraph. We proceed to list some minimal strict split obstructions. The family S of digraphs with vertices a, b, c , d consists of all digraphs with 1. 2. 3. 4. 5.
a a a a a
∼ b ∼ c ∼ d ∼ a, or ∼ b ∼ c ∼ d ∼ a, and a ÷ c, or ∼ b, c ∼ d, or ∼ b, c ∼ d, a ÷ c, or ∼ b, c ∼ d, a ÷ c , b ÷ c.
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Fig. 3. The family S of minimal strict split obstructions.
It is easy to verify that each digraph in S is a minimal strict split obstruction. These digraphs are depicted in Fig. 3. We also note that the digraphs in the family D from Fig. 1 are all minimal strict split obstructions. (Tournaments on three vertices and semi-complete digraphs on four vertices without a strong clique on three vertices that do not contain a subtournament on three vertices.) Finally, since the underlying graph of a strict split graph must be split and hence chordal, the symmetric pentagon a ↔ b ↔ c ↔ d ↔ e ↔ a is also a minimal strict split obstruction. It turns out that there are no other forbidden induced subgraphs: Theorem 4. A digraph D is a strict split digraph if and only if it does not contain any of the following induced subgraphs.
• a digraph from the family D , • a digraph on four vertices from the family S • or the symmetric pentagon. Lemma 5. A minimal strict split obstruction does not contain a strong clique on three vertices. Proof. For a contrapositive argument, let D be a digraph and {x, y, z } = X ⊆ V (D) such that the subgraph induced by X , D[X ], is a strong clique of D. We will prove by induction on the number of pairs of adjacent vertices in D that D is not a minimal strict split obstruction. If D has only three pairs of adjacent vertices, then (V (D) \ X , X ) is a split partition of D, hence D is not a strict split obstruction. If D has at least four pairs of adjacent vertices, then we consider u, v ∈ V (D) such that u ∼ v and u ̸∈ X . Notice that if D is a symmetric digraph, then the problem is equivalent to the undirected case, and, since D contains a triangle, it is not a minimal split obstruction. Thus, we can choose u and v such that a is an asymmetric arc between u and v in D. Let D′ = D − a. By induction hypothesis D′ is not a minimal strict split obstruction. If D′ is a split obstruction, then it contains a minimal strict split obstruction H. If u ̸∈ V (H ) or v ̸∈ V (H ), then H is an induced subgraph of D, and hence D is not a minimal strict split obstruction. Otherwise, either H + a is a minimal strict split obstruction, or it contains a minimal strict split obstruction as an induced subgraph, and hence D does too. This follows from the fact that H + a is not strict split unless H is strict split. So, suppose that D′ has a strict split partition (V0 , V1 ). Since u and v are not adjacent in D′ , then we can assume without loss of generality that u ∈ V0 . If v ∈ V1 , then (V0 , V1 ) is a strict split partition of D. Thus, v ∈ V0 . If u → V1 and V1 → u, then (V0 \ {u}, V1 ∪ {u}) is a split partition of D. Using an analogous argument for v , we derive the existence of vertices u1 , v1 ∈ V1 such that u ̸↔ u1 and v ̸↔ v1 . First, suppose that u1 ̸= v1 . If we let W = {u, v, u1 , v1 } be a subset of V (D), then it is clear that D[W ] contains neither a strong clique on 3 vertices, nor an independent set of size 3. If D[W ] is a strict split obstruction, then it properly contains an induced minimal strict split obstruction, and so does D. So, suppose that D[W ] is a strict split digraph, then u ↔ v1 and v ̸∼ u1 , or v ↔ u1 and u ̸∼ v1 . Since the cases are analogous, we will only consider the former. If u ↔ w for every w ∈ V1 \ {u1 }, and u1 ̸∼ z for every z ∈ V0 \ {u}, then swapping u and u1 in (V0 , V1 ) results in a strict split partition of D. Hence, there is a vertex w ∈ V (D) such that w ∈ V0 \ {u, v} and w ∼ u1 , or w ∈ V1 \ {u1 , v1 } and u ̸↔ w . In the former case, {u, v, u1 , w} induces a digraph on four vertices of the family S in D. In the latter case, if v ∼ w , then it is easy to verify that D[{u, v, v1 , w}] either contains a tournament on three vertices or belongs to S . If v is non-adjacent with w in D, then it is also easy to verify that D[{u, v, u1 , w}] is a digraph in S .
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If u1 = v1 , then u ↔ w and v ↔ w for every w ∈ V1 \ {u1 }. If both u and v are adjacent with u1 , then D[{u, v, u1 }] is a tournament on 3 vertices. Suppose that v is not adjacent with u1 . Thus, there exists a vertex w ∈ V0 \ {u, v} adjacent with u1 . Otherwise, swapping u and u1 in (V0 , V1 ) results in a strict split partition of D. Again, it is easy to verify that D[{u, v, w, u1 }] is a digraph of the family S . Since the outcome for every case is that D is not a minimal split obstruction, the desired result follows. Lemma 6. A minimal strict split obstruction does not contain a semi-complete digraph on five vertices. Proof. Suppose that a minimal split obstruction D has a semi-complete induced subgraph H on five vertices. If no arc of H is a digon, then it contains a tournament on three vertices, which is a minimal split obstruction, contradicting the minimality of D. Otherwise, let u, v ∈ V (H ) be such that u ↔ v . Since H has five vertices, there are three other vertices v1 , v2 , v3 in V (H ). If vi ↔ vj for some 1 ≤ i < j ≤ 3, then, since Lemma 5 guarantees that there are no symmetric triangles in D, D[{u, v, vi , vj }] is a digraph of the family S , and hence a minimal split obstruction. Therefore, we can assume that all such arcs are asymmetric. But again, D[{v1 , v2 , v3 }] is a tournament on three vertices. Proof of Theorem 4. Let D be a minimal strict split obstruction. It is routine to verify that the two tournaments on three vertices are the only minimal strict split obstructions on three vertices. Although tedious, it is also routine to verify that the only minimal strict split obstructions on four vertices are the elements of S . We will show that, except for the symmetric pentagon, every minimal strict split obstruction has at most four vertices. If the underlying graph of D is not a split graph, then it contains an induced copy of C4 , C4 , or C5 . In every case, it follows that D contains an induced bi-orientation of such subgraph. In the first two cases such bi-orientations are digraphs of the family S , and hence, D has exactly four vertices. If D contains an induced C5 , then we have two cases. If the pentagon is symmetric in D, then it is a minimal strict split obstruction, and it follows that D itself is a symmetric pentagon. Otherwise, there is at least one asymmetric arc in the pentagon, implying the existence of a digraph of the family S as an induced subgraph of D, contradicting that D is a minimal strict split obstruction. So, we can suppose that the underlying graph of D has a split partition (V0 , V1 ) maximizing the size of V1 . Since D is a strict split obstruction, V1 cannot be a strong clique of D, else, (V0 , V1 ) would be a strict split partition of D. Hence, |V1 | ≥ 2. Also, it follows from Lemma 6 that |V1 | ≤ 4. We will show that |V1 | = 2 implies |V (D)| = 4. If V1 = {u, v}, we can assume without loss of generality that u → v . If no vertex of V0 is adjacent with u, then (V0 ∪ {u}, {v}) is a strict split partition of D, a contradiction. Hence, there is at least one vertex u′ ∈ V0 adjacent with u. Analogously, there is a vertex v ′ ∈ V0 adjacent with v . If u′ = v ′ , then (V0 \ {u′ }, {u, v, u′ }) is a split partition of the underlying graph of D, which results in a contradiction because (V0 , V1 ) was chosen to maximize the size of V1 . Thus, u′ ̸= v ′ and D[{u, v, u′ , v ′ }] induces a digraph of the family S . Since every digraph in the family S is a minimal strict split obstruction, D = D[{u, v, u′ , v ′ }] and D has four vertices. If |V1 | = 4, since D is a minimal obstruction then D[V1 ] contains neither a tournament on three vertices nor a strong clique on three vertices. Hence D[V1 ] is a digraph of the family S and, since D is a minimal obstruction, D = D[V1 ]. Therefore |V (D)| = 4. Finally, if V1 = {x, y, z }, then we can assume without loss of generality that y ↔ z and x ÷ y. By the choice of (V0 , V1 ), there is no vertex in V0 adjacent with all vertices of V1 . If no vertex in V0 is adjacent with x, then (V0 ∪ {x}, {y, z }) is a strict split partition of D. Hence, there is w ∈ V0 such that w ∼ x. We will assume first, without loss of generality, that w ∼ y. Thus, z ̸∼ w and, since D does not contain tournaments on three vertices as subgraphs, it is easy to verify that D[{w, x, y, z }] is a digraph of the family S . We conclude that D = D[{w, x, y, z }] and it has exactly four vertices. So, suppose that w ̸∼ y, w ̸∼ z. If x ÷ z, then D[{w, x, y, z }] belongs to S . Hence, assume x ↔ z. If no vertex in V0 is adjacent to y, then (V0 ∪ {y}, {x, z }) is a strict split partition of D, a contradiction. Hence, a vertex v ∈ V0 exists such that v ̸= w and v ∼ y. If v ∼ x, then v ̸∼ z and D[{v, x, y, z }] is a digraph of the family S . Otherwise, D[{v, w, x, y}] belongs to the family S . In either case, the minimality of D is contradicted. Since the cases are exhaustive, we conclude that every minimal split obstruction digraph has at most four vertices, except for the symmetric pentagon. 4. Further results on strict split digraphs We first describe our ordering characterization of strict split digraphs. Theorem 7. A digraph D is a strict split digraph if and only if its vertices can be linearly ordered by < so that for any u < v < w , if u ∼ v then v ↔ w . Proof. If D is a strict split digraph with partition (V0 , V1 ), then such an ordering < is obtained by listing first all vertices of V0 (in any order), and then all vertices in V1 (in any order). On the other hand, given such an ordering < of the vertices of D, we can define a partition (V0 , V1 ) as follows. If D has no arcs we may set V1 = ∅. Otherwise, let z be the first vertex in the ordering < such that some x with x < z is adjacent with z. Then we may set V0 to consist of all vertices preceding z in <, and V1 to consist of z and all vertices following it in <. It is easy to see that the property of < ensures that this is a strict split partition of D.
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This is analogous to the ordering characterization of split graphs [10]. We also have the following result analogous to the situation for undirected split graphs. Corollary 8. D is a strict split digraph if and only if both D and D are strict chordal digraphs. Proof. If D is a strict split digraph, then by Theorem 7 its vertices can be linearly ordered by < so that for any u < v < w , if u ∼ v then v ↔ w . In particular, this ordering < satisfies the weaker requirement that for any u < v < w , if u ∼ v and u ∼ w then v ↔ w , so D is a strict chordal digraph. Moreover, the ordering <′ obtained by reversing <, satisfies for any x <′ y <′ z, if x ̸↔ y then y ̸∼ z in D, which is equivalent to x ∼ y implies y ↔ z in D, so D is also a strict chordal digraph. Whether a graph G is split depends solely by its degree sequence [17]. This is also true for strict split digraphs. It is interesting to observe that the condition is expressed in terms of the (total) degrees, rather than the in-degrees and outdegrees. (The degree of a vertex in a digraph is the sum of its in-degree and out-degree.) Theorem 9. Let the degree sequence of a digraph D be d1 ≥ · · · ≥ dn , and let m be the largest value of i such that di ≥ 2(i − 1). Then D is a strict split digraph if and only if m
di = 2m(m − 1) +
i=1
n
di .
i=m+1
If this is the case, then the m vertices with the largest degrees form a maximum strong clique in G, and the remaining vertices constitute an independent set. Proof. Let D be a digraph with the aforementioned property. Let X , Y be the sets X = {v1 , . . . , vm } and Y = {vm+1 , . . . , vn }. Observe that the number of arcs with both ends in X is given by a(X ) =
1 2
m
di − a(X , Y ) − a(Y , X )
i=1
=
1 2
2m(m − 1) +
n
di − (a(X , Y ) + a(Y , X )) .
i=m+1
Also, observe that i=m+1 di ≥ a(X , Y ) + a(Y , X ) with equality if and only if Y is an independent set. So, if Y is not an independent set, then we have the following strict inequality
n
a(X ) > m(m − 1), a contradiction because D is a simple digraph. Hence, Y must be an independent set of D, and thus a(X ) = m(m − 1). But this implies that X is a strong clique of D. Now, let D be a strict split digraph with degree sequence d1 ≥ · · · ≥ dn . Suppose that D has strict split partition (X , Y ) with X a strong clique of maximum cardinality m in D. It is clear that X = {v1 , . . . , vm }, or vm has a false twin in D, so we may interchange labels to obtain X = {v1 , . . . , vm }. Since X is a strong clique and Y is an independent set, it is not hard to observe that m
di = 2m(m − 1) +
i=1
n
di .
i=m+1
Also, vm is strongly adjacent with every vertex in {v1 , . . . , vm−1 }, thus dm ≥ 2(m − 1). But N + (vm+1 ), N − (vm+1 ) ⊆ {v1 , . . . , vm }, and X is a maximum strong clique of D, hence d(vm+1 ) < 2m. So, m is the largest value of i such that di ≥ 2(i − 1). The theorem implies an obvious O(m + n) recognition algorithm for split digraphs. 5. Concluding remarks We first discuss our claim that bi-orientations of chordal graphs are a small proportion of all strict chordal digraphs, and bi-orientations of split graphs a small proportion of all strict split digraphs. Indeed, the number of split graphs with a clique of k vertices and an independent set of ℓ vertices is 2kℓ , and hence this is also the number of their bi-orientations with k vertices in the strong clique and ℓ vertices in the independent set. On the other hand, the total number of strict split digraphs with these parameters is 4kℓ , so that the symmetric cases represent a vanishingly small proportion of all strict split digraphs. Since almost all chordal graphs are split [2], similar comments apply to strict chordal digraphs. Next we relate strict chordal digraphs to perfect digraphs studied in [1]. In fact, if D is a strict chordal digraph, then both D and its complement are perfect digraphs. Indeed, it is shown in [1] that a digraph D is perfect if and only if it does not contain an induced directed cycle, and its symmetric arcs form a perfect graph. The symmetric arcs of a strict chordal digraph form a chordal graph, thus a perfect graph. Moreover, by Theorem 1, strict chordal digraphs cannot contain an induced directed cycle. (Directed three-cycle is in the family D , and all other directed cycles are in the family C .) Therefore a strict chordal digraph is perfect. It is also shown in [1] that a digraph D is perfect if and only if the underlying graph G of its complement
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D is perfect, and D is clique-acyclic, i.e., contains no directed cycle (consisting of weak arcs oriented in the same direction) that induces a clique in G. Note that a strict chordal digraph D is clique-acyclic, because any clique in G induces a strict chordal subgraph of D and hence all but one vertex form a strong clique. Moreover, the underlying graph of D is chordal hence perfect. It follows from this that D is a perfect digraph as well. Thus we have three subclasses of perfect digraphs (strict chordal, complements of strict chordal, and strict split digraphs) with polynomial time recognition algorithms as well as polynomial time optimization algorithms for some basic combinatorial optimization problems, including finding a maximum acyclic subgraph, and finding the dichromatic number [1]. (These problems can be solved in polynomial time for all perfect digraphs, but the recognition of perfect digraphs is an N P -complete problem [1].) The authors of [1] ask for interesting subclasses of perfect digraphs. We have argued that strict chordal digraphs (and their complements) and strict split digraphs are such subclasses. We further illustrate the point by discussing kernels. A set of vertices N is a kernel of a digraph D if it is independent and every vertex not in N dominates some vertex of N [32]. It is shown in [1] that the complement of every perfect digraph has a kernel. Hence every strict chordal digraph has a kernel, and every complement of a strict chordal digraph has a kernel. By contrast, it is N P -complete to decide whether a perfect digraph has a kernel [1]. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]
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Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Strict chordal and strict split digraphs✩ Pavol Hell a,∗ , César Hernández-Cruz a,b a
School of Computing Science, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6
b
Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad Universitaria, C.P. 04510, México, D.F., Mexico
article
info
Article history: Received 14 April 2015 Received in revised form 9 February 2016 Accepted 11 February 2016 Available online xxxx
abstract We introduce new versions of chordal and split digraphs, and explore their similarity with the corresponding undirected notions. © 2016 Elsevier B.V. All rights reserved.
Keywords: Split digraph Chordal digraph Perfect digraph Recognition algorithm
1. Introduction A graph G is chordal if its vertices can be linearly ordered by < so that for any u < v < w , if u ∼ v and u ∼ w then v ∼ w . Such an ordering < is called a perfect elimination ordering of G. Chordal graphs are an important class of perfect graphs, admit elegant recognition algorithms and characterizations, efficient optimization algorithms, and interesting applications [17]. They arise, for example, from consideration of sparse symmetric matrices whose sparseness can be preserved during Gaussian elimination [34]. A graph is chordal if and only if it does not have an induced cycle of length greater than three [17]. Chordal graphs can be recognized in linear time [34]; in fact the algorithm can be made certifying, in the sense that it either finds a simplicial ordering or an induced cycle of length greater than three [35]. The chromatic number, as well as the size of a maximum clique and maximum independent set in a chordal graph can be found in linear time, from the perfect elimination ordering [15,17]. Even nicer properties hold for its subclass of split graphs. A graph is split if its vertex set can be partitioned into a clique and an independent set [13]. A graph G is split if and only if both G and G are chordal [17]. It can be seen from this that a graph is split if and only if it does not have an induced C4 , C5 , or C4 [13]. Further, a graph is split if and only if its vertices can be linearly ordered by < so that for any u < v < w , if u ∼ v then v ∼ w [10]. Whether or not an input graph is split can be recognized in linear time just by considering its degree sequence [17,13]. A certifying linear time recognition algorithm for split graphs is given in [24,23]. It is known that the proportion of chordal graphs on n vertices that are split graphs tends to one as n increases [2]. Perhaps the best known subclass is the class of interval graphs, since it has the most numerous natural applications [3,8]. A graph G is an interval graph if its vertices v can be represented by intervals Iv so that v ∼ w if and only if Iv intersects Iw [17]. A graph is an interval graph if and only if its vertices can be linearly ordered by < so that for any u < v < w , if u ∼ w
✩ This research was supported by a research grant from NSERC Canada and ERCCZ LL 1201.
∗
Corresponding author. E-mail addresses: [email protected] (P. Hell), [email protected] (C. Hernández-Cruz).
http://dx.doi.org/10.1016/j.dam.2016.02.009 0166-218X/© 2016 Elsevier B.V. All rights reserved.
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then v ∼ w [17]. (It follows that an interval graph is necessarily chordal.) A forbidden induced subgraph characterization appears in [28] (see also [16,14]), and there are several linear-time recognition algorithms [5,25,19,9], and certifying algorithms [26]. All these graph classes are contained in the class of perfect graphs. A graph G is perfect if the chromatic number and the maximum clique size are equal for G and all of its induced subgraphs. The class of perfect graphs is considered of prime importance in algorithmic graph theory [4,17,7], a prototype framework for combinatorial max–min results. A graph is perfect if and only if it does not contain an induced cycle of odd length greater than three, or its complement [4,7]. Perfect graphs can be recognized in polynomial time [6], and the basic optimization problems of finding the chromatic number, the clique number, and the independence number can be solved in polynomial time [18]. There have been attempts to translate the elegance of these results to the realm of digraphs. Perhaps the best known is the digraph analogue of interval graphs, defined as follows [11]. (A different definition was proposed in [20]; it only yields acyclic digraphs.) A digraph D is an interval digraph if its vertices v can be represented by pairs of intervals Iv , Jv , so that v → w if and only if Iv intersects Jw . Some characterizations of interval digraphs are known [11], especially in terms of their adjacency matrices; however there is no forbidden induced subgraph (or substructure) characterization known, and the existing polynomial time recognition algorithms have high-degree-polynomial time bounds [31]. (We note that a low-degree-polynomial time algorithm is claimed in [33].) Interestingly, there is a more convenient subclass of interval digraphs. In [12] the authors define a digraph D to be an adjusted interval digraph if its vertices v can be represented by pairs of intervals Iv , Jv , so that v → w if and only if Iv intersects Jw , where the intervals Iv and Jv have the same left endpoint (are ‘‘left-adjusted’’). It turns out that adjusted interval digraphs show a greater similarity to interval graphs than plain interval digraphs. In particular, they have a forbidden structure characterization, an ordering characterization, and a low-degree-polynomial time recognition algorithm [12]. It turned out that interval digraphs were too ambitious a generalization of interval graphs, and by being more modest (while still making a large digraph generalization of interval graphs), we obtain a better analogue. Chordal digraphs are defined in [30,22], also in a way to correspond to sparse general matrices (not necessarily symmetric as above), whose sparseness can be preserved during Gaussian elimination. (Again a different definition tailored to acyclic digraphs was proposed in [21].) Specifically, a digraph D is a chordal digraph if its vertices can be linearly ordered by < so that for any u < v < w , if v → u and u → w then v → w [30,22]. Some natural analogues of the results about chordal graphs are obtained for the class of these chordal digraphs [30]. However, a forbidden subgraph characterization is not known, except in certain very special cases [30,21,29]. Split digraphs were defined in [27] as follows. D is a split digraph if its vertices can be partitioned into four sets A, B, C , D where A is a strong clique (a complete symmetric digraph), D is an independent set, all possible arcs go from A to C and from B to A and C , and no arcs go from C to B or D and from D to B. Moreover, it is required that the partition does not place all vertices in B or all vertices in C . (Otherwise every digraph would be split.) The main appeal of these split digraphs seems to be that can be recognized by just considering their degree sequence, just as split graphs can [27]. However, this class of split digraphs is not closed under taking induced subgraphs; any digraph D can be made split by the addition of one new vertex dominating all vertices of D. It appears that these digraph analogues of chordal and split graphs may again be too general to recover the elegance of chordal and split graphs. In this paper, we propose two new notions of chordal and split digraphs, which we call strict chordal digraphs and strict split digraphs. These properly generalize the undirected cases, in the sense that a chordal (respectively split) graph, when viewed as a digraph, by replacing each edge uv by the two arcs uv and v u, is a strict chordal (respectively strict split) digraph. There is however a much greater number of strict chordal and strict split digraphs that do not arise this way from their undirected analogues (see the Concluding Remarks). Nevertheless, the underlying graph of any strict chordal digraph is a chordal graph, and the underlying graph of any strict split digraph is a split graph. Moreover our classes turn out to be close enough to the undirected notions to allow extending some of the fundamental theorems from the undirected case, including forbidden subgraph characterizations and polynomial time recognition algorithms (linear time in the case of strict split graphs). The class of perfect graphs also has a digraph analogue. For digraphs, D is a perfect digraph [1] if for D and every induced subgraph of D, the dichromatic number and the clique number are equal. (The dichromatic number of D is the smallest number of colours for which D can be coloured so that each colour induces an acyclic subgraph; and the clique number of D is the number of vertices in a largest strong clique.) The authors of [1] have succeeded in obtaining many results analogous to the undirected cases, including a forbidden subgraph characterization, and polynomial time optimization algorithms for some basic problems. However, the recognition of perfect digraphs is N P -complete [1]. It will turn out that strict chordal digraphs and (therefore) the strict split digraphs are perfect in this sense. As used above, in a graph G we write u ∼ v to mean there is an edge joining u and v in G; we also say that u and v are adjacent. Similarly, in a digraph D, u → v means there is an arc from u to v ; we also say that u dominates v . In addition to these standard terms we also use the following helpful terminology, in a digraph D. We say that u and v are adjacent in (a digraph) D if u → v or v → u (or both); in analogy with graphs we write u ∼ v in this case. We say that u and v are strongly adjacent in D if u → v and v → u; in this case we write u ↔ v . (We also call u ↔ v a digon or a symmetric arc.) We say that u and v are weakly adjacent in D if either u → v or v → u; in this case we write u ÷ v . (In the special case when u → v but not v → u we write u → v and similarly for v → u.)
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In the figures adjacency u ∼ v is depicted by a solid line joining u and v , strong adjacency u ↔ v by a solid line joining u and v and having arrows at both ends, and weak adjacency u ÷ v by a (slightly thinner) solid line joining u and v and marked by a short bar. A strong clique in a digraph D is a set of vertices in which any two distinct vertices are strongly adjacent; a tournament is a set of vertices in which any two distinct vertices are weakly adjacent; and a semi-complete digraph is a set of vertices in which any two distinct vertices are adjacent. Let D be a digraph and G the undirected graph on the same vertices with u ∼ v in G if and only if u ∼ v in D. We call G the underlying graph of D, and also call D a super-orientation of G. An orientation of G is a super-orientation without strongly adjacent vertices (digons). A bi-orientation of G is a super-orientation without weakly adjacent vertices (all arcs are digons). 2. Strict chordal digraphs Recall that a digraph is chordal [22,30] if its vertices can be linearly ordered by < so that for any u < v < w , if v → u and u → w then v → w . We propose the following variant of this notion. A digraph is strict chordal if its vertices can be linearly ordered by < so that for any u < v < w, if v ∼ u and u ∼ w then v ↔ w . In other words, if < orders the vertices as v1 , v2 , . . . , vn , then the set of vertices from vi+1 , vi+2 , . . . , vn that are adjacent with vi form a strong clique. In particular, v1 is adjacent with a set of vertices that form a strong clique; such a vertex is called simplicial. Note that strict chordal digraphs are a restricted class of chordal digraphs. Note also that, as suggested earlier, the biorientation of any chordal graph is a strict chordal digraph. In fact, it turns out that strict chordal digraphs are a non-trivial and interesting class of digraphs, properly more general than just bi-orientations of undirected chordal graphs, but still well enough behaved. In particular, we shall provide a forbidden induced structure characterization for them. First we note that there is a polynomial time recognition algorithm for strict chordal digraphs. Indeed, since an induced subgraph of a strict chordal digraph is again strict chordal, we can test whether or not D is a strict chordal digraph by finding a simplicial vertex v of D and testing recursively that D − v is strict chordal. A strict chordal obstruction is a digraph which is not a strict chordal digraph. A minimal strict chordal obstruction is a strict chordal obstruction D such that every proper induced subgraph of D is a strict chordal digraph. Observe that the underlying graph of a strict chordal digraph D must be chordal. Moreover, the graph formed by the symmetric edges (digons) of a strict chordal digraph D must be a chordal graph. In fact, the ordering < from the definition of a strict chordal digraph D is a perfect elimination ordering of both the underlying graph and the graph formed by the symmetric edges of D. We now list some basic minimal strict chordal obstructions. It follows from the last remarks that any super-orientation of a cycle of length greater than three is a minimal strict chordal obstruction. Moreover, any super-orientation of the four-cycle C4 with a chord that is not a digon is a strict chordal obstruction. Indeed, the first vertex in any ordering < cannot be incident to the chord, since the same ordering must be a perfect elimination ordering for the underlying graph; thus the first vertex is one of the other two vertices, and consequently the chord should have been a digon. It is easy to see that it is in fact a minimal strict chordal obstruction. Therefore we denote by C the family of minimal strict chordal obstructions consisting of all super-orientations of cycles of length greater than three, together with all super-orientations of C4 with a chord that is not a digon. (See the upper part of Fig. 1; the conventions concerning the various kinds of lines depicting the arcs are detailed at the end of the previous section.) It is easy to see that the definition of < implies that any tournament on at least three vertices is a strict chordal obstruction. (In an ordering < the second and third vertex need to be strongly adjacent.) Therefore the two three-vertex tournaments (the directed three-cycle and the transitive triple) are minimal strict chordal obstructions. Similarly, any semi-complete digraph with n vertices that is a strict chordal digraph must contain a strong clique containing n − 1 vertices (excluding the first vertex in the ordering <). Thus any four-vertex semi-complete digraph without a strong clique on three vertices is a strict chordal obstruction. Let D denote the family consisting of both tournaments on three vertices and all four-vertex semi-complete digraphs without a strong clique on three vertices. (The lower part of Fig. 1 depicts all minimal members of D , up to symmetry. Note that several four-vertex digraphs are not minimal because they contain a tournament on three vertices.) There are two other infinite families of forbidden induced subgraphs. The lollipop on vertices a0 , a1 , a2 , a3 , . . . , ak , k ≥ 2, has arcs a0 ↔ a1 , a0 ÷ a2 , a1 ÷ a2 , and a2 ∼ a3 ∼ a4 · · · ∼ ak . The vertex ak is called the end-vertex of the lollipop. The sum of a lollipop on a0 , a1 , a2 , a3 , . . . , ak and a lollipop on b0 , b1 , b2 , b3 , . . . , bℓ is obtained from their disjoint union by identifying ak = bℓ . The join of these two lollipops is obtained from their disjoint union by the addition of all digons ak ↔ bi , i = 0, 1, 2, . . . , ℓ and bℓ ↔ aj , j = 0, 1, 2, . . . , k. The family L consists of all sums of lollipops, and the family K consists of all joins of lollipops, see Fig. 2. It turns out that families C , D , L, and K contain all the minimal obstructions to strict chordal digraphs. We proceed to prove our forbidden induced subgraph characterization of strict chordal digraphs. Theorem 1. D is a strict chordal digraph if and only if it does not contain an induced subgraph from C ∪ D ∪ L ∪ K . Proof. We have already shown all digraphs in C ∪ D ∪ L ∪ K are strict chordal obstructions. Therefore, let D be a digraph without induced subgraphs from the families C , D , L, and K ; we shall prove that D is strict chordal. Note that the underlying graph of D is chordal, since D contains no induced subgraph from C .
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Fig. 1. The minimal strict chordal obstruction families C and D .
Fig. 2. The minimal obstruction families L and K built of lollipops.
Lemma 2. If D does not contain an induced subgraph from C ∪ D ∪ L ∪ K and is not semi-complete, then any minimal cutset of D is a strong clique. Proof. Let S be a minimal cutset of D. Since the underlying graph of D is chordal, it follows from a corresponding result on chordal graphs [17] that S is a semi-complete digraph. If S is not a strong clique, then it contains weakly adjacent vertices u and v . Since S is a minimal cutset of D, there are components D1 and D2 of D − S such that u and v are adjacent with vertices in both D1 and D2 . Let Pi be a shortest uv -path all whose internal vertices lie in Di , i ∈ {1, 2}. Then either Pi ∪ {u, v} induces a chordless cycle for some i ∈ {1, 2}, or P1 ∪ P2 ∪ (u, v) induces a four-cycle with a chord that is not a digon; both are in the family C , a contradiction. Note that a strict chordal digraph always contains a simplicial vertex (the first vertex in the ordering <). Lemma 3. If a strict chordal digraph D does not contain two non-adjacent simplicial vertices, then D is semi-complete, or every simplicial vertex of D is the end-vertex of an induced lollipop.
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Proof. We will proceed by induction on the number of vertices of D. The result is easy to verify for strict chordal digraphs D with fewer than five vertices. Let D be a non-semi-complete connected chordal digraph on n ≥ 5 vertices. Let x be a simplicial vertex of D and let y be a vertex non-adjacent with x. By Lemma 2, a minimal cutset S separating x from y must be a strong clique. Let X and Y be the components of D − S containing x and y, respectively. Every simplicial vertex of D[Y ∪ S ] belongs to S, otherwise D would have two non-adjacent simplicial vertices. Let u be a simplicial vertex of D[Y ∪ S ]. Suppose first that D[Y ∪ S ] is semi-complete. Since S is a strong clique, there are u1 , u2 ∈ Y \ S such that u ÷ u1 and u ÷ u2 (otherwise some vertex in Y would be a simplicial vertex of D). Now, if P is an induced xu-path in the underlying graph of D, then V (P ) ∪ {u1 , u2 } induces a lollipop in D with x as its end-vertex. If D[Y ∪ S ] is not semi-complete, then, by the induction hypothesis, it contains an induced lollipop with end vertex u ∈ S. Again, if P is an induced xu-path in the underlying graph of D, and Z are the vertices of the induced lollipop of D[Y ∪ S ], then V (P ) ∪ Z induces a lollipop in D with end-vertex x. We note that for semi-complete digraphs D, if there is a unique arc that is not a digon then the two vertices on the arc are both simplicial. Otherwise, if there are at least two arcs that are not digons, there is at most one simplicial vertex. We are now ready to show, by induction on the number of vertices of D, that every C ∪ D ∪ L ∪ K -free digraph has a simplicial vertex. The result is again easy to verify for fewer than five vertices. Let D be a C ∪ D ∪ L ∪ K -free digraph with n vertices, n ≥ 5. If D is semicomplete, since it is D -free, it must contain a strong clique on n − 1 vertices as an induced subgraph. Observe that the remaining vertex is a simplicial vertex of D. Otherwise, let x and y be non-adjacent vertices in D, and let S be a minimal vertex cut separating x from y. If X and Y are the components of D − S containing x and y, respectively, then D1 = D[X ∪ S ] and D2 = D[Y ∪ S ] are strict chordal digraphs. If D1 or D2 contains two non-adjacent simplicial vertices, then D contains a simplicial vertex. So, suppose that every simplicial vertex of each D1 and D2 belongs to S. We claim that this cannot happen. Let s1 and s2 be simplicial vertices in D1 and D2 , respectively. There are three cases. In the first case, both D1 and D2 are semi-complete digraphs. Let u1 , u2 and v1 , v2 be the vertices weakly adjacent with s1 and s2 , respectively. Hence, {s1 , s2 , u1 , u2 , v1 , v2 } induces a digraph from L if s1 = s2 , or a digraph from K otherwise. In the second case, assume that D1 is semi-complete and D2 is not. If s1 = s2 , then, D contains an induced subgraph isomorphic to an element of L. Otherwise, let H be the lollipop on a0 , . . . , ak = s2 induced in D2 . Suppose that s1 ∼ a0 . Since D is C -free, s1 ↔ ai for every 0 ≤ i ≤ k. Let u1 , u2 ∈ X be such that s1 ÷ u1 and s1 ÷ u2 . Clearly, V (H ) ∪ {s1 , u1 , u2 } induces in D a digraph from K . Therefore, s1 is non-adjacent with a0 and a1 . But this implies that D contains an element of L as an induced subgraph. In the third case, neither D1 nor D2 is semi-complete. Let H1 be the lollipop on a0 , . . . , ak = s1 induced in D1 , and let H2 be the lollipop on b0 , . . . , bl = s2 induced in D2 . If s1 = s2 , then D[H1 ∪ H2 ] belongs to L. Else, if s1 is not adjacent with b0 or s2 is not adjacent with a0 , then H1 ∪ H2 contains an induced copy of a digraph in L. Assume that s1 is adjacent with b0 and s2 is adjacent with a0 . Since D is C -free, s1 ↔ bi for 0 ≤ i ≤ l, and s2 ↔ ai for 0 ≤ i ≤ k. Hence, D[H1 ∪ H2 ] belongs to K . Therefore at least one of D1 , D2 must have a simplicial vertex not contained in S. Hence, D has a simplicial vertex. By repeating the argument recursively we obtain the desired ordering <, proving that D is a strict chordal digraph. 3. Strict split digraphs A digraph D is a strict split graph if its vertices can be partitioned into a strong clique and an independent set. Such a partition will be called a strict split partition of D. Although the class of strict split digraphs is smaller than the class of split digraphs introduced in [27], it contains all biorientations of split graphs, and has interesting properties comparable to the class of split graphs. In particular, it admits a characterization by a finite set of forbidden induced subgraphs, an ordering characterization, and is the intersection of the class of strict chordal digraphs and the class of their complements. It is also the case that the underlying graph of a strict split digraph is a split graph. A strict split obstruction is a digraph which is not a strict split digraph. A minimal strict split obstruction is a strict split obstruction such that every proper induced subgraph is a strict split digraph. We proceed to list some minimal strict split obstructions. The family S of digraphs with vertices a, b, c , d consists of all digraphs with 1. 2. 3. 4. 5.
a a a a a
∼ b ∼ c ∼ d ∼ a, or ∼ b ∼ c ∼ d ∼ a, and a ÷ c, or ∼ b, c ∼ d, or ∼ b, c ∼ d, a ÷ c, or ∼ b, c ∼ d, a ÷ c , b ÷ c.
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Fig. 3. The family S of minimal strict split obstructions.
It is easy to verify that each digraph in S is a minimal strict split obstruction. These digraphs are depicted in Fig. 3. We also note that the digraphs in the family D from Fig. 1 are all minimal strict split obstructions. (Tournaments on three vertices and semi-complete digraphs on four vertices without a strong clique on three vertices that do not contain a subtournament on three vertices.) Finally, since the underlying graph of a strict split graph must be split and hence chordal, the symmetric pentagon a ↔ b ↔ c ↔ d ↔ e ↔ a is also a minimal strict split obstruction. It turns out that there are no other forbidden induced subgraphs: Theorem 4. A digraph D is a strict split digraph if and only if it does not contain any of the following induced subgraphs.
• a digraph from the family D , • a digraph on four vertices from the family S • or the symmetric pentagon. Lemma 5. A minimal strict split obstruction does not contain a strong clique on three vertices. Proof. For a contrapositive argument, let D be a digraph and {x, y, z } = X ⊆ V (D) such that the subgraph induced by X , D[X ], is a strong clique of D. We will prove by induction on the number of pairs of adjacent vertices in D that D is not a minimal strict split obstruction. If D has only three pairs of adjacent vertices, then (V (D) \ X , X ) is a split partition of D, hence D is not a strict split obstruction. If D has at least four pairs of adjacent vertices, then we consider u, v ∈ V (D) such that u ∼ v and u ̸∈ X . Notice that if D is a symmetric digraph, then the problem is equivalent to the undirected case, and, since D contains a triangle, it is not a minimal split obstruction. Thus, we can choose u and v such that a is an asymmetric arc between u and v in D. Let D′ = D − a. By induction hypothesis D′ is not a minimal strict split obstruction. If D′ is a split obstruction, then it contains a minimal strict split obstruction H. If u ̸∈ V (H ) or v ̸∈ V (H ), then H is an induced subgraph of D, and hence D is not a minimal strict split obstruction. Otherwise, either H + a is a minimal strict split obstruction, or it contains a minimal strict split obstruction as an induced subgraph, and hence D does too. This follows from the fact that H + a is not strict split unless H is strict split. So, suppose that D′ has a strict split partition (V0 , V1 ). Since u and v are not adjacent in D′ , then we can assume without loss of generality that u ∈ V0 . If v ∈ V1 , then (V0 , V1 ) is a strict split partition of D. Thus, v ∈ V0 . If u → V1 and V1 → u, then (V0 \ {u}, V1 ∪ {u}) is a split partition of D. Using an analogous argument for v , we derive the existence of vertices u1 , v1 ∈ V1 such that u ̸↔ u1 and v ̸↔ v1 . First, suppose that u1 ̸= v1 . If we let W = {u, v, u1 , v1 } be a subset of V (D), then it is clear that D[W ] contains neither a strong clique on 3 vertices, nor an independent set of size 3. If D[W ] is a strict split obstruction, then it properly contains an induced minimal strict split obstruction, and so does D. So, suppose that D[W ] is a strict split digraph, then u ↔ v1 and v ̸∼ u1 , or v ↔ u1 and u ̸∼ v1 . Since the cases are analogous, we will only consider the former. If u ↔ w for every w ∈ V1 \ {u1 }, and u1 ̸∼ z for every z ∈ V0 \ {u}, then swapping u and u1 in (V0 , V1 ) results in a strict split partition of D. Hence, there is a vertex w ∈ V (D) such that w ∈ V0 \ {u, v} and w ∼ u1 , or w ∈ V1 \ {u1 , v1 } and u ̸↔ w . In the former case, {u, v, u1 , w} induces a digraph on four vertices of the family S in D. In the latter case, if v ∼ w , then it is easy to verify that D[{u, v, v1 , w}] either contains a tournament on three vertices or belongs to S . If v is non-adjacent with w in D, then it is also easy to verify that D[{u, v, u1 , w}] is a digraph in S .
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If u1 = v1 , then u ↔ w and v ↔ w for every w ∈ V1 \ {u1 }. If both u and v are adjacent with u1 , then D[{u, v, u1 }] is a tournament on 3 vertices. Suppose that v is not adjacent with u1 . Thus, there exists a vertex w ∈ V0 \ {u, v} adjacent with u1 . Otherwise, swapping u and u1 in (V0 , V1 ) results in a strict split partition of D. Again, it is easy to verify that D[{u, v, w, u1 }] is a digraph of the family S . Since the outcome for every case is that D is not a minimal split obstruction, the desired result follows. Lemma 6. A minimal strict split obstruction does not contain a semi-complete digraph on five vertices. Proof. Suppose that a minimal split obstruction D has a semi-complete induced subgraph H on five vertices. If no arc of H is a digon, then it contains a tournament on three vertices, which is a minimal split obstruction, contradicting the minimality of D. Otherwise, let u, v ∈ V (H ) be such that u ↔ v . Since H has five vertices, there are three other vertices v1 , v2 , v3 in V (H ). If vi ↔ vj for some 1 ≤ i < j ≤ 3, then, since Lemma 5 guarantees that there are no symmetric triangles in D, D[{u, v, vi , vj }] is a digraph of the family S , and hence a minimal split obstruction. Therefore, we can assume that all such arcs are asymmetric. But again, D[{v1 , v2 , v3 }] is a tournament on three vertices. Proof of Theorem 4. Let D be a minimal strict split obstruction. It is routine to verify that the two tournaments on three vertices are the only minimal strict split obstructions on three vertices. Although tedious, it is also routine to verify that the only minimal strict split obstructions on four vertices are the elements of S . We will show that, except for the symmetric pentagon, every minimal strict split obstruction has at most four vertices. If the underlying graph of D is not a split graph, then it contains an induced copy of C4 , C4 , or C5 . In every case, it follows that D contains an induced bi-orientation of such subgraph. In the first two cases such bi-orientations are digraphs of the family S , and hence, D has exactly four vertices. If D contains an induced C5 , then we have two cases. If the pentagon is symmetric in D, then it is a minimal strict split obstruction, and it follows that D itself is a symmetric pentagon. Otherwise, there is at least one asymmetric arc in the pentagon, implying the existence of a digraph of the family S as an induced subgraph of D, contradicting that D is a minimal strict split obstruction. So, we can suppose that the underlying graph of D has a split partition (V0 , V1 ) maximizing the size of V1 . Since D is a strict split obstruction, V1 cannot be a strong clique of D, else, (V0 , V1 ) would be a strict split partition of D. Hence, |V1 | ≥ 2. Also, it follows from Lemma 6 that |V1 | ≤ 4. We will show that |V1 | = 2 implies |V (D)| = 4. If V1 = {u, v}, we can assume without loss of generality that u → v . If no vertex of V0 is adjacent with u, then (V0 ∪ {u}, {v}) is a strict split partition of D, a contradiction. Hence, there is at least one vertex u′ ∈ V0 adjacent with u. Analogously, there is a vertex v ′ ∈ V0 adjacent with v . If u′ = v ′ , then (V0 \ {u′ }, {u, v, u′ }) is a split partition of the underlying graph of D, which results in a contradiction because (V0 , V1 ) was chosen to maximize the size of V1 . Thus, u′ ̸= v ′ and D[{u, v, u′ , v ′ }] induces a digraph of the family S . Since every digraph in the family S is a minimal strict split obstruction, D = D[{u, v, u′ , v ′ }] and D has four vertices. If |V1 | = 4, since D is a minimal obstruction then D[V1 ] contains neither a tournament on three vertices nor a strong clique on three vertices. Hence D[V1 ] is a digraph of the family S and, since D is a minimal obstruction, D = D[V1 ]. Therefore |V (D)| = 4. Finally, if V1 = {x, y, z }, then we can assume without loss of generality that y ↔ z and x ÷ y. By the choice of (V0 , V1 ), there is no vertex in V0 adjacent with all vertices of V1 . If no vertex in V0 is adjacent with x, then (V0 ∪ {x}, {y, z }) is a strict split partition of D. Hence, there is w ∈ V0 such that w ∼ x. We will assume first, without loss of generality, that w ∼ y. Thus, z ̸∼ w and, since D does not contain tournaments on three vertices as subgraphs, it is easy to verify that D[{w, x, y, z }] is a digraph of the family S . We conclude that D = D[{w, x, y, z }] and it has exactly four vertices. So, suppose that w ̸∼ y, w ̸∼ z. If x ÷ z, then D[{w, x, y, z }] belongs to S . Hence, assume x ↔ z. If no vertex in V0 is adjacent to y, then (V0 ∪ {y}, {x, z }) is a strict split partition of D, a contradiction. Hence, a vertex v ∈ V0 exists such that v ̸= w and v ∼ y. If v ∼ x, then v ̸∼ z and D[{v, x, y, z }] is a digraph of the family S . Otherwise, D[{v, w, x, y}] belongs to the family S . In either case, the minimality of D is contradicted. Since the cases are exhaustive, we conclude that every minimal split obstruction digraph has at most four vertices, except for the symmetric pentagon. 4. Further results on strict split digraphs We first describe our ordering characterization of strict split digraphs. Theorem 7. A digraph D is a strict split digraph if and only if its vertices can be linearly ordered by < so that for any u < v < w , if u ∼ v then v ↔ w . Proof. If D is a strict split digraph with partition (V0 , V1 ), then such an ordering < is obtained by listing first all vertices of V0 (in any order), and then all vertices in V1 (in any order). On the other hand, given such an ordering < of the vertices of D, we can define a partition (V0 , V1 ) as follows. If D has no arcs we may set V1 = ∅. Otherwise, let z be the first vertex in the ordering < such that some x with x < z is adjacent with z. Then we may set V0 to consist of all vertices preceding z in <, and V1 to consist of z and all vertices following it in <. It is easy to see that the property of < ensures that this is a strict split partition of D.
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This is analogous to the ordering characterization of split graphs [10]. We also have the following result analogous to the situation for undirected split graphs. Corollary 8. D is a strict split digraph if and only if both D and D are strict chordal digraphs. Proof. If D is a strict split digraph, then by Theorem 7 its vertices can be linearly ordered by < so that for any u < v < w , if u ∼ v then v ↔ w . In particular, this ordering < satisfies the weaker requirement that for any u < v < w , if u ∼ v and u ∼ w then v ↔ w , so D is a strict chordal digraph. Moreover, the ordering <′ obtained by reversing <, satisfies for any x <′ y <′ z, if x ̸↔ y then y ̸∼ z in D, which is equivalent to x ∼ y implies y ↔ z in D, so D is also a strict chordal digraph. Whether a graph G is split depends solely by its degree sequence [17]. This is also true for strict split digraphs. It is interesting to observe that the condition is expressed in terms of the (total) degrees, rather than the in-degrees and outdegrees. (The degree of a vertex in a digraph is the sum of its in-degree and out-degree.) Theorem 9. Let the degree sequence of a digraph D be d1 ≥ · · · ≥ dn , and let m be the largest value of i such that di ≥ 2(i − 1). Then D is a strict split digraph if and only if m
di = 2m(m − 1) +
i=1
n
di .
i=m+1
If this is the case, then the m vertices with the largest degrees form a maximum strong clique in G, and the remaining vertices constitute an independent set. Proof. Let D be a digraph with the aforementioned property. Let X , Y be the sets X = {v1 , . . . , vm } and Y = {vm+1 , . . . , vn }. Observe that the number of arcs with both ends in X is given by a(X ) =
1 2
m
di − a(X , Y ) − a(Y , X )
i=1
=
1 2
2m(m − 1) +
n
di − (a(X , Y ) + a(Y , X )) .
i=m+1
Also, observe that i=m+1 di ≥ a(X , Y ) + a(Y , X ) with equality if and only if Y is an independent set. So, if Y is not an independent set, then we have the following strict inequality
n
a(X ) > m(m − 1), a contradiction because D is a simple digraph. Hence, Y must be an independent set of D, and thus a(X ) = m(m − 1). But this implies that X is a strong clique of D. Now, let D be a strict split digraph with degree sequence d1 ≥ · · · ≥ dn . Suppose that D has strict split partition (X , Y ) with X a strong clique of maximum cardinality m in D. It is clear that X = {v1 , . . . , vm }, or vm has a false twin in D, so we may interchange labels to obtain X = {v1 , . . . , vm }. Since X is a strong clique and Y is an independent set, it is not hard to observe that m
di = 2m(m − 1) +
i=1
n
di .
i=m+1
Also, vm is strongly adjacent with every vertex in {v1 , . . . , vm−1 }, thus dm ≥ 2(m − 1). But N + (vm+1 ), N − (vm+1 ) ⊆ {v1 , . . . , vm }, and X is a maximum strong clique of D, hence d(vm+1 ) < 2m. So, m is the largest value of i such that di ≥ 2(i − 1). The theorem implies an obvious O(m + n) recognition algorithm for split digraphs. 5. Concluding remarks We first discuss our claim that bi-orientations of chordal graphs are a small proportion of all strict chordal digraphs, and bi-orientations of split graphs a small proportion of all strict split digraphs. Indeed, the number of split graphs with a clique of k vertices and an independent set of ℓ vertices is 2kℓ , and hence this is also the number of their bi-orientations with k vertices in the strong clique and ℓ vertices in the independent set. On the other hand, the total number of strict split digraphs with these parameters is 4kℓ , so that the symmetric cases represent a vanishingly small proportion of all strict split digraphs. Since almost all chordal graphs are split [2], similar comments apply to strict chordal digraphs. Next we relate strict chordal digraphs to perfect digraphs studied in [1]. In fact, if D is a strict chordal digraph, then both D and its complement are perfect digraphs. Indeed, it is shown in [1] that a digraph D is perfect if and only if it does not contain an induced directed cycle, and its symmetric arcs form a perfect graph. The symmetric arcs of a strict chordal digraph form a chordal graph, thus a perfect graph. Moreover, by Theorem 1, strict chordal digraphs cannot contain an induced directed cycle. (Directed three-cycle is in the family D , and all other directed cycles are in the family C .) Therefore a strict chordal digraph is perfect. It is also shown in [1] that a digraph D is perfect if and only if the underlying graph G of its complement
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D is perfect, and D is clique-acyclic, i.e., contains no directed cycle (consisting of weak arcs oriented in the same direction) that induces a clique in G. Note that a strict chordal digraph D is clique-acyclic, because any clique in G induces a strict chordal subgraph of D and hence all but one vertex form a strong clique. Moreover, the underlying graph of D is chordal hence perfect. It follows from this that D is a perfect digraph as well. Thus we have three subclasses of perfect digraphs (strict chordal, complements of strict chordal, and strict split digraphs) with polynomial time recognition algorithms as well as polynomial time optimization algorithms for some basic combinatorial optimization problems, including finding a maximum acyclic subgraph, and finding the dichromatic number [1]. (These problems can be solved in polynomial time for all perfect digraphs, but the recognition of perfect digraphs is an N P -complete problem [1].) The authors of [1] ask for interesting subclasses of perfect digraphs. We have argued that strict chordal digraphs (and their complements) and strict split digraphs are such subclasses. We further illustrate the point by discussing kernels. A set of vertices N is a kernel of a digraph D if it is independent and every vertex not in N dominates some vertex of N [32]. It is shown in [1] that the complement of every perfect digraph has a kernel. Hence every strict chordal digraph has a kernel, and every complement of a strict chordal digraph has a kernel. By contrast, it is N P -complete to decide whether a perfect digraph has a kernel [1]. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]
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