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New graph classes characterized by weak vertex separators and two-pairs
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New graph classes characterized by weak vertex separators and two-pairs Terry A. McKee Department of Mathematics & Statistics, Wright State University, Dayton, OH 45435, USA Received 29 August 2015; accepted 30 September 2016 Available online xxxx
Abstract A set of vertices whose deletion from a graph would increase the distance between two remaining vertices is called a weak vertex separator of the graph. Two vertices form a two-pair if all chordless paths between them have length 2. I prove that every inclusion-minimal weak vertex separator of every induced subgraph of a graph G induces a complete subgraph if and only if nonadjacent vertices of induced C4 subgraphs of G always form two-pairs of G; moreover, this is also true when “complete” and C4 are replaced with “edgeless” and K 4 − e (∼ =K 1,1,2 ). The first of the resulting new graph classes generalizes chordal graphs, and the second generalizes unichord-free graphs; they both generalize distance-hereditary graphs and geodetic graphs. c 2016 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND ⃝ license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Two-pair; Minimal vertex separator; Minimal weak separator; Induced-minimal weak separator; Unichord-free
1. Inclusion-minimal separators and two-pairs If x and y are vertices in a connected graph G, then S ⊆ V (G) − {x, y} is a minimal x, y-separator of G if x and y are in different components of the induced subgraph G − S and yet x and y are in a common component of G − S ′ for all proper subsets S ′ of S. A minimal x, y-separator is also called a minimal vertex separator, or minimal separator for short. The graph on the left in Fig. 1 has minimal separators {2, 3}, {3, 4, 5}, and {5, 6}. A connected graph has no minimal separators if and only if it is complete. As an example of terminological awkwardness, a minimal separator can be properly contained in another minimal separator. For instance, in the graph on the right in Fig. 1, {6} is a minimal 7, 8-separator that is properly contained in the minimal 4, 7-separator {5, 6}. Following the terminology of [1], an inclusion-minimal separator is a minimal separator that does not properly contain another minimal separator. Thus {2, 3}, {3, 4, 5}, and {6} are the only inclusion-minimal separators in the graph on the right in Fig. 1.
Peer review under responsibility of Kalasalingam University. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.akcej.2016.11.008 c 2016 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license 0972-8600/⃝ (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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Fig. 1. Two graphs to illustrate minimal separators.
Fig. 2. From left to right, the house, domino, and gem graphs.
A chordal graph is a graph in which every cycle of length 4 or more has a chord (an edge between two nonconsecutive vertices of the cycle). Chordal graphs are also characterized by the property that every minimal separator induces a complete subgraph; see [2,3]. A unichord-free graph is a graph in which no cycle of length 4 or more has a unique chord. These were introduced in [4] as the graphs in which every minimal separator induces an edgeless subgraph and, independently, in [5] from a constructive point of view that addresses computation issues. These graphs have been further studied – and given the “unichord-free” name – in several recent papers [6–9]. A weakly chordal graph (sometimes called a weakly triangulated graph) is a graph G such that, in both G and its complement G, every cycle of length 5 or more has a chord; see [2,10,3]. A two-pair of a graph is a pair of nonadjacent vertices such that every chordless path between them has length 2; see [2]. Two-pairs were invented in [11] (although they were also important, yet unnamed, in [12]) to characterize weakly chordal graphs by the property that every minimal separator induces a complete subgraph or contains a two-pair. A distance-hereditary graph is a graph G such that, for every connected induced subgraph H of G and every x, y ∈ V (H ), the x-to-y distance d H (x, y) in H equals dG (x, y); see [2]. Distance-hereditary graphs were characterized in [12] by the property that they contain no chordless cycle Cn with n ≥ 5 and no induced house, domino, or gem subgraph (see Fig. 2), and also by the property that the nonadjacent vertices of an induced path P3 (∼ =K 1,2 ) always form a two-pair. A geodetic graph is a graph in which every two vertices are connected by a unique shortest path; see [2]. Section 2 will introduce minimal weak separators and inclusion-minimal weak separators. Sections 3 and 4 will use inclusion-minimal weak separators to define two new classes of graphs that generalize, respectively, the classes of chordal graphs and unichord-free graphs. Each of these new classes will also have a simple two-pair characterization. 2. Inclusion-minimal weak separators If x and y are vertices in a connected graph G, then S ⊂ V (G) − {x, y} is a minimal weak x, y-separator of G if x and y are in a common component of G − S with dG−S (x, y) > dG (x, y) and yet dG−S ′ (x, y) = dG (x, y) for all proper subsets S ′ of S. A minimal weak x, y-separator is also called a minimal weak vertex separator, or minimal weak separator for short. As examples, each vertex v of Cn forms a minimal weak separator {v} when n ≥ 5, while C4 has no minimal weak separators. The two graphs in Fig. 1 have minimal weak separators {3} and {5} and {3, 5}. A connected graph has no minimal weak separators if and only if it is distance-hereditary [13]. The graphs in Fig. 1 show that a minimal weak separator can be properly contained in another minimal weak separator; for instance, {3} is a minimal weak 1, 6-separator, and {3, 5} is a minimal weak 1, 7-separator. Motivated by the terminology of [1], define an inclusion-minimal weak separator S to be a minimal weak separator that does not properly contain another minimal weak separator. In the graphs in Fig. 1, {3} and {5} are the only inclusion-minimal weak separators. Minimal separators have proved useful in graph theory, as evidenced by their role in [2,4,3], while inclusionminimal separators have not (at least partly for the reasons mentioned in [1]). And yet inclusion-minimal weak
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separators – instead of minimal weak separators – will be important in Sections 3 and 4 (corresponding results in terms of minimal weak separators would seem to be considerably more complicated). Ref. [13] discusses minimal weak separators of chordal graphs. (WARNING: [13, Thm. 4] and [13, Cor. 6] need to be corrected so as to apply only to inclusion-minimal weak separators; Fig. 1 gives counterexamples to using arbitrary minimal weak separators. The proofs of [13, Thm. 4] and [13, Cor. 6] then need “inclusion-” in front of each mention of “minimal weak separator”, as do the statement of [3, Exer. 2.9] and the proof of [13, Thm. 5], although the latter theorem does happen to be true for arbitrary minimal weak separators, as will follow from Lemma 1.) Lemma 1. The following are equivalent for every vertex v of a graph G: (1a) v is contained in a minimal weak separator of G. (1b) v is a vertex of a cycle C of length 5 or more such that v is an endpoint of every chord of C. (1c) v is a vertex of a chordless Cn with n ≥ 5 or is a maximum-degree vertex of an induced house, domino, or gem subgraph of G. (1d) G contains a chordless x-to-y path of length 3 or more with v ∈ N (x) ∩ N (y). Proof. First suppose v is in a minimal weak x, y-separator S of G, and assume further that x, y, and S have been chosen so that dG (x, y) is minimum. Since S is a weak x, y-separator containing v, there is at least one chordless x-to-y path π through v that has length dG (x, y) and there is at least one x-to-y path τ that is internally-disjoint from π and has length greater than dG (x, y); among all such paths τ , assume that τ has been chosen to have minimum length. Let C be the cycle π ∪ τ , so C has length 5 or more and contains v. If ab is a chord of C with (say) a in the x-to-v subpath of π , then b ∈ V (τ ) and a, b ∈ R ∪ S − {x} where R is the component of G − S that contains x; thus a = v, to prevent a and y from contradicting the assumed minimality of dG (x, y) (if a ̸= v, then the a-to-y subpath π ′ of π and the a-to-y path τ ′ formed by appending the edge ab to the b-to-y subpath of τ would have lengths |π ′ | < |π | and |π ′ | < |τ ′ | ≤ |τ |). Therefore, every possible chord of C has endpoint v. Conversely, suppose v and C are as in (1b). Let x and y be the two neighbors of v along C, so dG (x, y) = 2 and E(C) − {vx, vy} is a chordless x-to-y path π that has length |V (C)| − 2 ≥ 3. By (1b), V (π ) is disjoint from S = N (x) ∩ N (y), and so S is a minimal weak x, y-separator of G that contains v. Finally, the equivalences of (1b) with (1c) and (1d) are straightforward to check. The following three known results follow from Lemma 1: (i) if G is chordal, then a vertex v is in a minimal weak separator of G if and only if v is the degree-4 vertex of an induced gem subgraph; (ii) if G is distance-hereditary, then no vertex is in a minimal weak separator of G; (iii) if G is geodetic, then every vertex in a minimal weak separator of G is in a chordless cycle of G of length 5 or more. Lemma 2. If v and w are vertices in an inclusion-minimal weak separator, then there exist vertices x and y and a chordless x-to-y path of length 3 or more such that v, w ∈ N (x) ∩ N (y). Proof. Suppose v and w are in an inclusion-minimal weak separator S of a graph G. Because S is a minimal weak separator, Lemma 1 implies that v is in a cycle C of length 5 or more such that v is an endpoint of every chord of C. Let x and y be the neighbors of v along C. The x-to-y path C − v is a chordless x-to-y path with length 3 or more. Because the weak separator S ̸= {v} is inclusion-minimal, {v} is not a minimal weak separator of G and so, in particular, {v} is not a weak x, y-separator of G. Thus, the shortest x-to-y paths in G −v still have length 2, and the set S ′ = N (x) ∩ N (y) is a minimal weak x, y-separator and has S ′ ⊆ S. Since the weak separator S is inclusion-minimal, S ′ = S and so w ∈ S ′. Thus, the shortest x-to-w-to-y path in G − v has length 2, and so v, w ∈ N (x) ∩ N (y). Taking vertices v = 3 and w = 5 in the graphs of Fig. 1 illustrates the necessity of requiring an inclusion-minimal weak separator in Lemma 2 (since {3, 5} is a minimal weak separator that is not inclusion-minimal). 3. Complete inclusion-minimal weak separators If G is a chordal, distance-hereditary, or geodetic graph, then every inclusion-minimal weak separator S of G induces a complete subgraph of G (since S is contained in a minimal separator and so induces a complete subgraph of a chordal graph, since distance-hereditary graphs have no weak separators at all, and since every minimal weak separator of a geodetic graph is a singleton). All the inclusion-minimal weak separators of the graphs in Figs. 1 and 2
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Fig. 3. A graph in which the nonadjacent vertices z and z ′ form a minimal weak x, y-separator that is an inclusion-minimal weak separator.
Fig. 4. A graph illustrating condition (3a) of Theorem 3.
also induce complete subgraphs, but this fails for the graph in Fig. 3 (each of the “solid” vertices constitutes an inclusion-minimal weak separator, as does {z, z ′ }). Theorem 3 will characterize the graphs for which every inclusion-minimal weak separator of every induced subgraph induces a complete subgraph by requiring certain 4-cycles – namely, those in condition (3b) that have x, v, y, w in that order – to have a chord (namely, vw) and by a two-pair characterization. Theorem 3. The following are equivalent for every graph: (3a) Every inclusion-minimal weak separator of every induced subgraph induces a complete subgraph. (3b) For every chordless x-to-y path of length 3 or more, every two vertices v, w ∈ N (x) ∩ N (y) are adjacent. (3c) Nonadjacent vertices of induced C4 subgraphs always form two-pairs of the graph. Proof. First suppose a graph G satisfies (3a) and contains a chordless x-to-y path π of length 3 or more and vertices v, w ∈ N (x) ∩ N (y). Let H be the subgraph of G induced by V (π ) ∪ {v, w}. Since {v, w} is an inclusion-minimal weak separator of H , condition (3a) implies that v and w are adjacent. Thus, (3b) holds. Conversely, suppose (3a) fails with H an induced subgraph G and S an inclusion-minimal weak separator of H where S contains nonadjacent vertices v and w. By Lemma 2, H contains a chordless x-to-y path of length 3 or more with the nonadjacent vertices v, w ∈ N (x) ∩ N (y). Thus, (3b) would fail. Finally, the equivalence of (3b) with (3c) is straightforward to check. In the graph G in Fig. 4, each of the “solid” vertices constitutes an inclusion-minimal weak separator. Thus, every inclusion-minimal weak separator of G trivially induces a complete graph, but (3b) and (3c) fail. This shows the necessity of the phrase “of every induced subgraph” in the statement of (3a), since the graph in Fig. 3 is an induced subgraph of G that makes (3a) fail. Similarly, {z, z ′ } is a minimal – but not inclusion-minimal – weak x, y-separator of G, and every minimal weak separator of G induces a complete graph, showing the necessity of “inclusion-minimal” in the statement of (3a). Corollary 4. If a graph has no induced C4 subgraph, then every inclusion-minimal weak separator of every induced subgraph induces a complete subgraph. Corollary 4 follows from condition (3c) holding vacuously whenever there is no induced C4 . 4. Edgeless inclusion-minimal weak separators If G is a unichord-free, distance-hereditary, or geodetic graph, then every inclusion-minimal weak separator S of G induces an edgeless subgraph of G (as at the beginning of Section 3). All the inclusion-minimal weak separators of the graphs in Figs. 1, 2 and 3 also induce edgeless subgraphs, but this fails for the graph obtained from Fig. 3
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by inserting an edge zz ′ (each of the “solid” vertices again constitutes an inclusion-minimal weak separator, as does {z, z ′ }). Theorem 5 will characterize the graphs for which every inclusion-minimal weak separator of every induced subgraph induces an edgeless subgraph by requiring certain 4-cycles – namely, those in condition (5b) that have x, v, y, w in that order – to be chordless and by a two-pair characterization. Theorem 5. The following are equivalent for every graph: (5a) Every inclusion-minimal weak separator of every induced subgraph induces an edgeless subgraph. (5b) For every chordless x-to-y path of length 3 or more, every two vertices v, w ∈ N (x) ∩ N (y) are nonadjacent. (5c) Nonadjacent vertices of induced K 4 − e subgraphs always form two-pairs of the graph. Proof. The argument parallels the proof of Theorem 3, except switching adjacency of v and w with their nonadjacency. Corollary 6. If a graph has no induced K 4 − e subgraph, then every inclusion-minimal weak separator of every induced subgraph induces an edgeless subgraph. Corollary 6 follows from condition (5c) holding vacuously whenever there is no induced K 4 − e (these are commonly called diamond-free graphs and are characterized as the strictly clique irreducible graphs in [14]). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
T. Kloks, Treewidth, Constructions and Approximations, in: Lect. Notes Comput. Sci., vol. 842, Springer, Berlin, 1994. A. Brandst¨adt, V.-B. Le, J.P. Spinrad, Graph Classes; A Survey, Society for Industrial and Applied Mathematics, Philadelphia, 1999. T.A. McKee, F.R. McMorris, Topics in Intersection Graph Theory, Society for Industrial and Applied Mathematics, Philadelphia, 1999. T.A. McKee, Independent separator graphs, Util. Math. 73 (2007) 217–224. N. Trotignon, K. Vuˇskovi´c, A structure theorem for graphs with no cycle with a unique chord and its consequences, J. Graph Theory 63 (2010) 31–67. R.C.S. Machado, C.M.H. de Figueiredo, Total chromatic number of unichord-free graphs, Discrete Appl. Math. 159 (2011) 1851–1864. R.C.S. Machado, C.M.H. de Figueiredo, N. Trotignon, Complexity of colouring problems restricted to unichord-free and {square,unichord}free graphs, Discrete Appl. Math. 164 (2014) 191–199. R.C.S. Machado, C.M.H. de Figueiredo, K. Vuˇskovi´c, Chromatic index of graphs with no cycle with unique chord, Theoret. Comput. Sci. 411 (2010) 1221–1234. T.A. McKee, A new characterization of unichord-free graphs, Discuss. Math. Graph Theory 35 (2015) 765–771. R. Hayward, Weakly triangulated graphs, J. Combin. Theory Ser. B 39 (1985) 200–208. R. Hayward, C. Ho`ang, F. Maffray, Optimizing weakly triangulated graphs, Graphs Combin. 5 (1989) 339–349; Graphs Combin. 6 (1990) 33–35 (erratum). H.-J. Bandelt, H.M. Mulder, Distance-hereditary graphs, J. Combin. Theory Ser. B 41 (1986) 182–208. T.A. McKee, Minimal weak separators of chordal graphs, Ars Combin. 101 (2011) 321–331. W.D. Wallis, G.-H. Zhang, On maximal clique irreducible graphs, J. Combin. Math. Combin. Comput. 8 (1990) 187–193.
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New graph classes characterized by weak vertex separators and two-pairs Terry A. McKee Department of Mathematics & Statistics, Wright State University, Dayton, OH 45435, USA Received 29 August 2015; accepted 30 September 2016 Available online xxxx
Abstract A set of vertices whose deletion from a graph would increase the distance between two remaining vertices is called a weak vertex separator of the graph. Two vertices form a two-pair if all chordless paths between them have length 2. I prove that every inclusion-minimal weak vertex separator of every induced subgraph of a graph G induces a complete subgraph if and only if nonadjacent vertices of induced C4 subgraphs of G always form two-pairs of G; moreover, this is also true when “complete” and C4 are replaced with “edgeless” and K 4 − e (∼ =K 1,1,2 ). The first of the resulting new graph classes generalizes chordal graphs, and the second generalizes unichord-free graphs; they both generalize distance-hereditary graphs and geodetic graphs. c 2016 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND ⃝ license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Two-pair; Minimal vertex separator; Minimal weak separator; Induced-minimal weak separator; Unichord-free
1. Inclusion-minimal separators and two-pairs If x and y are vertices in a connected graph G, then S ⊆ V (G) − {x, y} is a minimal x, y-separator of G if x and y are in different components of the induced subgraph G − S and yet x and y are in a common component of G − S ′ for all proper subsets S ′ of S. A minimal x, y-separator is also called a minimal vertex separator, or minimal separator for short. The graph on the left in Fig. 1 has minimal separators {2, 3}, {3, 4, 5}, and {5, 6}. A connected graph has no minimal separators if and only if it is complete. As an example of terminological awkwardness, a minimal separator can be properly contained in another minimal separator. For instance, in the graph on the right in Fig. 1, {6} is a minimal 7, 8-separator that is properly contained in the minimal 4, 7-separator {5, 6}. Following the terminology of [1], an inclusion-minimal separator is a minimal separator that does not properly contain another minimal separator. Thus {2, 3}, {3, 4, 5}, and {6} are the only inclusion-minimal separators in the graph on the right in Fig. 1.
Peer review under responsibility of Kalasalingam University. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.akcej.2016.11.008 c 2016 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license 0972-8600/⃝ (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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Fig. 1. Two graphs to illustrate minimal separators.
Fig. 2. From left to right, the house, domino, and gem graphs.
A chordal graph is a graph in which every cycle of length 4 or more has a chord (an edge between two nonconsecutive vertices of the cycle). Chordal graphs are also characterized by the property that every minimal separator induces a complete subgraph; see [2,3]. A unichord-free graph is a graph in which no cycle of length 4 or more has a unique chord. These were introduced in [4] as the graphs in which every minimal separator induces an edgeless subgraph and, independently, in [5] from a constructive point of view that addresses computation issues. These graphs have been further studied – and given the “unichord-free” name – in several recent papers [6–9]. A weakly chordal graph (sometimes called a weakly triangulated graph) is a graph G such that, in both G and its complement G, every cycle of length 5 or more has a chord; see [2,10,3]. A two-pair of a graph is a pair of nonadjacent vertices such that every chordless path between them has length 2; see [2]. Two-pairs were invented in [11] (although they were also important, yet unnamed, in [12]) to characterize weakly chordal graphs by the property that every minimal separator induces a complete subgraph or contains a two-pair. A distance-hereditary graph is a graph G such that, for every connected induced subgraph H of G and every x, y ∈ V (H ), the x-to-y distance d H (x, y) in H equals dG (x, y); see [2]. Distance-hereditary graphs were characterized in [12] by the property that they contain no chordless cycle Cn with n ≥ 5 and no induced house, domino, or gem subgraph (see Fig. 2), and also by the property that the nonadjacent vertices of an induced path P3 (∼ =K 1,2 ) always form a two-pair. A geodetic graph is a graph in which every two vertices are connected by a unique shortest path; see [2]. Section 2 will introduce minimal weak separators and inclusion-minimal weak separators. Sections 3 and 4 will use inclusion-minimal weak separators to define two new classes of graphs that generalize, respectively, the classes of chordal graphs and unichord-free graphs. Each of these new classes will also have a simple two-pair characterization. 2. Inclusion-minimal weak separators If x and y are vertices in a connected graph G, then S ⊂ V (G) − {x, y} is a minimal weak x, y-separator of G if x and y are in a common component of G − S with dG−S (x, y) > dG (x, y) and yet dG−S ′ (x, y) = dG (x, y) for all proper subsets S ′ of S. A minimal weak x, y-separator is also called a minimal weak vertex separator, or minimal weak separator for short. As examples, each vertex v of Cn forms a minimal weak separator {v} when n ≥ 5, while C4 has no minimal weak separators. The two graphs in Fig. 1 have minimal weak separators {3} and {5} and {3, 5}. A connected graph has no minimal weak separators if and only if it is distance-hereditary [13]. The graphs in Fig. 1 show that a minimal weak separator can be properly contained in another minimal weak separator; for instance, {3} is a minimal weak 1, 6-separator, and {3, 5} is a minimal weak 1, 7-separator. Motivated by the terminology of [1], define an inclusion-minimal weak separator S to be a minimal weak separator that does not properly contain another minimal weak separator. In the graphs in Fig. 1, {3} and {5} are the only inclusion-minimal weak separators. Minimal separators have proved useful in graph theory, as evidenced by their role in [2,4,3], while inclusionminimal separators have not (at least partly for the reasons mentioned in [1]). And yet inclusion-minimal weak
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separators – instead of minimal weak separators – will be important in Sections 3 and 4 (corresponding results in terms of minimal weak separators would seem to be considerably more complicated). Ref. [13] discusses minimal weak separators of chordal graphs. (WARNING: [13, Thm. 4] and [13, Cor. 6] need to be corrected so as to apply only to inclusion-minimal weak separators; Fig. 1 gives counterexamples to using arbitrary minimal weak separators. The proofs of [13, Thm. 4] and [13, Cor. 6] then need “inclusion-” in front of each mention of “minimal weak separator”, as do the statement of [3, Exer. 2.9] and the proof of [13, Thm. 5], although the latter theorem does happen to be true for arbitrary minimal weak separators, as will follow from Lemma 1.) Lemma 1. The following are equivalent for every vertex v of a graph G: (1a) v is contained in a minimal weak separator of G. (1b) v is a vertex of a cycle C of length 5 or more such that v is an endpoint of every chord of C. (1c) v is a vertex of a chordless Cn with n ≥ 5 or is a maximum-degree vertex of an induced house, domino, or gem subgraph of G. (1d) G contains a chordless x-to-y path of length 3 or more with v ∈ N (x) ∩ N (y). Proof. First suppose v is in a minimal weak x, y-separator S of G, and assume further that x, y, and S have been chosen so that dG (x, y) is minimum. Since S is a weak x, y-separator containing v, there is at least one chordless x-to-y path π through v that has length dG (x, y) and there is at least one x-to-y path τ that is internally-disjoint from π and has length greater than dG (x, y); among all such paths τ , assume that τ has been chosen to have minimum length. Let C be the cycle π ∪ τ , so C has length 5 or more and contains v. If ab is a chord of C with (say) a in the x-to-v subpath of π , then b ∈ V (τ ) and a, b ∈ R ∪ S − {x} where R is the component of G − S that contains x; thus a = v, to prevent a and y from contradicting the assumed minimality of dG (x, y) (if a ̸= v, then the a-to-y subpath π ′ of π and the a-to-y path τ ′ formed by appending the edge ab to the b-to-y subpath of τ would have lengths |π ′ | < |π | and |π ′ | < |τ ′ | ≤ |τ |). Therefore, every possible chord of C has endpoint v. Conversely, suppose v and C are as in (1b). Let x and y be the two neighbors of v along C, so dG (x, y) = 2 and E(C) − {vx, vy} is a chordless x-to-y path π that has length |V (C)| − 2 ≥ 3. By (1b), V (π ) is disjoint from S = N (x) ∩ N (y), and so S is a minimal weak x, y-separator of G that contains v. Finally, the equivalences of (1b) with (1c) and (1d) are straightforward to check. The following three known results follow from Lemma 1: (i) if G is chordal, then a vertex v is in a minimal weak separator of G if and only if v is the degree-4 vertex of an induced gem subgraph; (ii) if G is distance-hereditary, then no vertex is in a minimal weak separator of G; (iii) if G is geodetic, then every vertex in a minimal weak separator of G is in a chordless cycle of G of length 5 or more. Lemma 2. If v and w are vertices in an inclusion-minimal weak separator, then there exist vertices x and y and a chordless x-to-y path of length 3 or more such that v, w ∈ N (x) ∩ N (y). Proof. Suppose v and w are in an inclusion-minimal weak separator S of a graph G. Because S is a minimal weak separator, Lemma 1 implies that v is in a cycle C of length 5 or more such that v is an endpoint of every chord of C. Let x and y be the neighbors of v along C. The x-to-y path C − v is a chordless x-to-y path with length 3 or more. Because the weak separator S ̸= {v} is inclusion-minimal, {v} is not a minimal weak separator of G and so, in particular, {v} is not a weak x, y-separator of G. Thus, the shortest x-to-y paths in G −v still have length 2, and the set S ′ = N (x) ∩ N (y) is a minimal weak x, y-separator and has S ′ ⊆ S. Since the weak separator S is inclusion-minimal, S ′ = S and so w ∈ S ′. Thus, the shortest x-to-w-to-y path in G − v has length 2, and so v, w ∈ N (x) ∩ N (y). Taking vertices v = 3 and w = 5 in the graphs of Fig. 1 illustrates the necessity of requiring an inclusion-minimal weak separator in Lemma 2 (since {3, 5} is a minimal weak separator that is not inclusion-minimal). 3. Complete inclusion-minimal weak separators If G is a chordal, distance-hereditary, or geodetic graph, then every inclusion-minimal weak separator S of G induces a complete subgraph of G (since S is contained in a minimal separator and so induces a complete subgraph of a chordal graph, since distance-hereditary graphs have no weak separators at all, and since every minimal weak separator of a geodetic graph is a singleton). All the inclusion-minimal weak separators of the graphs in Figs. 1 and 2
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Fig. 3. A graph in which the nonadjacent vertices z and z ′ form a minimal weak x, y-separator that is an inclusion-minimal weak separator.
Fig. 4. A graph illustrating condition (3a) of Theorem 3.
also induce complete subgraphs, but this fails for the graph in Fig. 3 (each of the “solid” vertices constitutes an inclusion-minimal weak separator, as does {z, z ′ }). Theorem 3 will characterize the graphs for which every inclusion-minimal weak separator of every induced subgraph induces a complete subgraph by requiring certain 4-cycles – namely, those in condition (3b) that have x, v, y, w in that order – to have a chord (namely, vw) and by a two-pair characterization. Theorem 3. The following are equivalent for every graph: (3a) Every inclusion-minimal weak separator of every induced subgraph induces a complete subgraph. (3b) For every chordless x-to-y path of length 3 or more, every two vertices v, w ∈ N (x) ∩ N (y) are adjacent. (3c) Nonadjacent vertices of induced C4 subgraphs always form two-pairs of the graph. Proof. First suppose a graph G satisfies (3a) and contains a chordless x-to-y path π of length 3 or more and vertices v, w ∈ N (x) ∩ N (y). Let H be the subgraph of G induced by V (π ) ∪ {v, w}. Since {v, w} is an inclusion-minimal weak separator of H , condition (3a) implies that v and w are adjacent. Thus, (3b) holds. Conversely, suppose (3a) fails with H an induced subgraph G and S an inclusion-minimal weak separator of H where S contains nonadjacent vertices v and w. By Lemma 2, H contains a chordless x-to-y path of length 3 or more with the nonadjacent vertices v, w ∈ N (x) ∩ N (y). Thus, (3b) would fail. Finally, the equivalence of (3b) with (3c) is straightforward to check. In the graph G in Fig. 4, each of the “solid” vertices constitutes an inclusion-minimal weak separator. Thus, every inclusion-minimal weak separator of G trivially induces a complete graph, but (3b) and (3c) fail. This shows the necessity of the phrase “of every induced subgraph” in the statement of (3a), since the graph in Fig. 3 is an induced subgraph of G that makes (3a) fail. Similarly, {z, z ′ } is a minimal – but not inclusion-minimal – weak x, y-separator of G, and every minimal weak separator of G induces a complete graph, showing the necessity of “inclusion-minimal” in the statement of (3a). Corollary 4. If a graph has no induced C4 subgraph, then every inclusion-minimal weak separator of every induced subgraph induces a complete subgraph. Corollary 4 follows from condition (3c) holding vacuously whenever there is no induced C4 . 4. Edgeless inclusion-minimal weak separators If G is a unichord-free, distance-hereditary, or geodetic graph, then every inclusion-minimal weak separator S of G induces an edgeless subgraph of G (as at the beginning of Section 3). All the inclusion-minimal weak separators of the graphs in Figs. 1, 2 and 3 also induce edgeless subgraphs, but this fails for the graph obtained from Fig. 3
T.A. McKee / AKCE International Journal of Graphs and Combinatorics (
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by inserting an edge zz ′ (each of the “solid” vertices again constitutes an inclusion-minimal weak separator, as does {z, z ′ }). Theorem 5 will characterize the graphs for which every inclusion-minimal weak separator of every induced subgraph induces an edgeless subgraph by requiring certain 4-cycles – namely, those in condition (5b) that have x, v, y, w in that order – to be chordless and by a two-pair characterization. Theorem 5. The following are equivalent for every graph: (5a) Every inclusion-minimal weak separator of every induced subgraph induces an edgeless subgraph. (5b) For every chordless x-to-y path of length 3 or more, every two vertices v, w ∈ N (x) ∩ N (y) are nonadjacent. (5c) Nonadjacent vertices of induced K 4 − e subgraphs always form two-pairs of the graph. Proof. The argument parallels the proof of Theorem 3, except switching adjacency of v and w with their nonadjacency. Corollary 6. If a graph has no induced K 4 − e subgraph, then every inclusion-minimal weak separator of every induced subgraph induces an edgeless subgraph. Corollary 6 follows from condition (5c) holding vacuously whenever there is no induced K 4 − e (these are commonly called diamond-free graphs and are characterized as the strictly clique irreducible graphs in [14]). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
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