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Domination in Graphoidally Covered Graphs: Least-Kernel Graphoidal Covers
Available online at www.sciencedirect.com
Electronic Notes in Discrete Mathematics 53 (2016) 433–444 www.elsevier.com/locate/endm
Domination in Graphoidally Covered Graphs: Least-Kernel Graphoidal Covers Purnima Gupta 1 Department of Mathematics Sri Venkateswara College University of Delhi, Delhi, India
Rajesh Singh 3,2 Department Of Mathematics University Of Delhi, Delhi, India
(Dedicated to the sweet memories of Dr. B.D Acharya)
Abstract Given a graph G = (V, E) (not necessarily finite), a graphoidal cover of G means a collection Ψ of non-trivial paths in G called Ψ-edges, which are not necessarily open (not necessarily finite), such that every vertex of G is an internal vertex of at most one path in Ψ and every edge of G is in exactly one path in Ψ. The set of all graphoidal covers of a graph G = (V, E) is denoted by GG and for a given Ψ ∈ GG the ordered pair (G, Ψ) is called a graphoidally covered graph. Two vertices u and v of G are Ψ-adjacent if they are the ends of an open Ψ-edge. A set D of vertices in (G, Ψ) is Ψ-independent if no two vertices in D are Ψ-adjacent and is said to be Ψ-dominating if every vertex of G is either in D or is Ψ-adjacent to a vertex in D; G is γΨ (G)-definable (γiΨ (G)-definable) if (G, Ψ) has a finite Ψ-dominating (Ψ-independent Ψ-dominating) set. Clearly, if G is γiΨ (G)-definable, then G is γΨ (G)-definable and γΨ (G) ≤ γiΨ (G). Further if for a graphoidal cover Ψ of G, γΨ (G) = γiΨ (G) then we call Ψ as a least-kernel graphoidal cover of G (in short, http://dx.doi.org/10.1016/j.endm.2016.05.037 1571-0653/© 2016 Elsevier B.V. All rights reserved.
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an LKG cover of G). A natural question arises: “Does every graph possess an LKG cover?” We firstly exhibit that not every graph possesses an LKG cover and thereafter, in the quest to find graphs possessing an LKG cover, we proved that every finite tree and every finite unicyclic graph admits an LKG cover. We further extend Allan Laskar theorem to infinite graphs by showing every γ(G)-definable infinite claw free graph has an LKG cover. Keywords: Graphoidal Cover, Graphoidal Domination, Independent Domination.
1
Introduction
Throughout this paper we will consider simple graphs without loops as treated in most of the standard text-books on graph theory such as [10]. Further, unless mentioned otherwise, graphs will be assumed to be connected and infinite. In [3], Acharya and Sampathkumar introduced the notion of a graphoidal cover and graphoidal covering number of a finite graph and since then a lot of work has been done and hundreds of paper have come up in this short span. In 1999, Acharya and Gupta [1] extended the concept of graphoidal covers to infinite graphs and introduced the notion of domination to discrete structures, called graphoidally covered graphs. Given a graph G = (V, E) (not necessarily finite), a graphoidal cover [1] of G means a collection Ψ of non-trivial paths in G (i.e, having length at least one) called Ψ-edges, which are not necessarily open (not necessarily finite), such that the following conditions are satisfied: GC1 Every vertex of G is an internal vertex of at most one path in Ψ. GC2 Every edge of G is in exactly one path in Ψ The set of all graphoidal covers of a graph G is denoted by GG and for a given Ψ ∈ GG , the ordered pair (G, Ψ) is called a graphoidally covered graph. It may be noted that a Ψ-edge could possibly be infinite; in particular, it may be a one-way infinite path (or, the often so-called ‘ray’) or a two-way infinite path. Further, a finite open Ψ-edge has two distinct end vertices and 1
Email: [email protected] (corresponding author) Email: singh [email protected] 3 The author is thankful to Indian organization CSIR-UGC for providing the research grant to carry out the research. 2
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a closed Ψ-edge has only one end vertex, which is specified by Ψ. Next, a one-way infinite Ψ-edge has exactly one end vertex while a two-way infinite Ψ-edge has no end vertex. It may also be noted that there are three types of vertices in a graphoidally covered graph (G, Ψ), viz. (a) a vertex which is not an internal vertex of any Ψ-edge, called a Ψ-exterior vertex or a black vertex and shown as a small filled circle in the diagrammatic representation of (G, Ψ), (b) a vertex which is not an end-vertex of any Ψ-edge, called a Ψ-interior vertex or a white vertex and shown as a small unfilled circle in the diagrammatic representation of (G, Ψ), and (c) a vertex which is neither an exterior vertex nor an interior vertex of (G, Ψ), which means it is an internal vertex to a Ψ-edge as also an end-vertex to at least one other Ψ-edge, called a Ψ-composite vertex – in the diagrammatic representation, this vertex is represented as an unfilled circle with as many small tangents to the circle as the number of Ψ-edges for which this vertex is the end-vertex and each tangent indicates an end-vertex of the corresponding Ψ-edge. In Fig. 1 we give diagrammatic representation of four different graphoidal covers of K4 namely, E, Ψ1 , Ψ2 , Ψ3 where E is the edge set of G, Ψ1 = {bd, bca, badc}, Ψ2 = {bd, ac, badcb} and Ψ3 = {badc, acbd}. In Fig. 2 we have a graphoidal cover of an infinite tree in which every Ψ-edge is a one way infinite path.
Fig. 1. Different graphoidal covers of complete graph K4 .
Clearly, E ∈ GG and hence we call E =: E(G), the edge set of G, the trivial graphoidal cover of G; a graphoidal cover that is not trivial will be referred to as being nontrivial. Two distinct vertices u and v of G are Ψ-adjacent if they are the ends of an open Ψ-edge; a vertex is said to be self Ψ-adjacent if it is both the ‘initial’ and the ‘terminal’ end of a closed Ψ-edge (i.e., a cycle
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Fig. 2. An infinite tree, each of the Ψ-edges is a one way infinite path
in Ψ) – in either case, we represent the fact by writing uΨv. The Ψ-degree dΨ (u) of a vertex u is then defined as the number of open Ψ-edges having u as their end-vertex. A vertex is said to be a Ψ-pendant vertex if its Ψ-degree is one and a vertex is said to be a Ψ-support if it is Ψ-adjacent to at least one Ψ-pendant vertex. A set D of vertices in a graphoidally covered graph (G, Ψ) is a Ψ-dominating set (or ‘Ψ-domset’ for short) of G if every vertex of G is either in D or is Ψadjacent to a vertex in D; if (G, Ψ) contains a finite Ψ-dominating set, then the least cardinality of such a set is denoted as γΨ (G) and is called the Ψdomination number of (G, Ψ) ; such graphs G are called γΨ (G)-definable. A set D of vertices in a graphoidally covered graph (G, Ψ) is a Ψ-independent set if no pair of distinct vertices in D are Ψ-adjacent and a Ψ-dominating set D, which is also Ψ-independent is called a Ψ-kernel. If (G, Ψ) contains a finite Ψ-kernel then the least cardinality of a Ψ-kernel of (G, Ψ) is denoted by γiΨ (G) and then G is said to be ‘γiΨ (G)-definable’. In case (G, Ψ) is γΨ (G)- (γiΨ (G)) definable then it is often convenient to call a Ψ-dominating set (Ψ-kernel) consisting of γΨ (G) (γiΨ (G)) vertices a γΨ (G)-set (γiΨ (G)-set). Clearly, if (G, Ψ) is γiΨ (G)-definable then it is γΨ (G)-definable, but the converse may not be true for infinite graphs. In Fig. 3, we have an infinite graph G and a graphoidal cover Ψ of G such that G is γΨ (G)-definable but not γiΨ (G)definable. However, if G is an infinite claw free graph, then for the trivial graphoidal cover Ψ = E, γΨ (G)-definability implies γiΨ (G)-definability and in
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that case γΨ (G) = γiΨ (G). This fact can be proved using the technique used in the proof of Allan-Laskar Theorem [5] on claw free graphs.
Fig. 3. (G, Ψ), where Ψ = {uv} ∪ {ua2 a1 , ua3 a2 , ...} ∪ {vb2 b1 , vb3 b2 , ...}, is γΨ (G)-definable but not γiΨ (G)-definable and γΨ (G) = {u, v}.
For Ψ = E(G) we denote γE(G) (G) =: γ(G), and call it the domination number of G, and denote γiE(G) (G) =: γi (G), and call it the independent domination number of G, subject to the existence of these parameters, well in compatibility with these notations in the theory of domination in the case of finite graphs (e.g., see [8,9,7]).
2
Least Kernel Graphoidal Graphs
Given a graph G and a graphoidal cover Ψ of G such that G is γiΨ (G)-definable then obviously γΨ (G) ≤ γiΨ (G). (1) For the trivial graphoidal cover Ψ = E, the above inequality reduces to γE (G) ≤ γiE (G)
i.e.,
γ(G) ≤ γi (G).
We call a graphoidal cover Ψ ∈ GG to be a least kernel graphoidal cover (or simply an LKG cover ) of G if G is γiΨ (G)-definable and γΨ (G) = γiΨ (G).
(2)
For Ψ = E(G), this reduces to the well known equality γ(G) = γi (G), first considered by Allan-Laskar. Since by Allan Laskar Theorem [5], for every finite claw free graph γ(G) = γi (G), therefore the trivial graphoidal cover of every finite claw free graph is an LKG cover.
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In Fig. 4, we illustrate a graph G and a non-trivial graphoidal cover Ψ of G such that equality holds in (1) i.e., Ψ is an LKG cover of G. Since for the infinite graphoidally covered graph (G, Ψ) in Fig. 3, γΨ (G) = 2 and γiΨ (G) is not defined, therefore Ψ is not an LKG cover.
Fig. 4. {c, b1 , b2 } is a γψ1 (G)(γiψ1 (G)) set and ψ1 is an LKG Cover
In Fig. 5, we give an LKG cover of the infinite graph shown in Fig. 3. The set {u, v} forms a γΨ (G)-set as well as γiΨ (G)-set, where u is Ψ-adjacent to all ui ’s and v is Ψ-adjacent to all vi ’s.
Fig. 5. Ψ = {vuu0 v0 , vv2 v1 , vv3 v2 , ...} ∪ {uu1 u0 , uu2 u1 , ...} is an LKG cover of the infinite graph G.
Now before moving ahead, let us look at some of the basic but important observations that will prove handy while exploring least kernel graphoidal covers. We put them in a series of lemmas. Lemma 2.1 Every pendant vertex of a graph G is a Ψ-pendant vertex, for each Ψ ∈ GG .
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Lemma 2.2 If u ∈ V (G) is a support to n(≥ 3) pendant vertices, then for each Ψ ∈ GG , at least (n − 2) of these pendant vertices are Ψ-pendant vertices, with u as their common Ψ-support. Lemma 2.3 Let Ψ ∈ GG be such that G is γiΨ (G)-definable and if u ∈ V (G) is a Ψ-support to more than one Ψ-pendant vertices, then every γΨ (G)-set of G contains u. Lemma 2.4 If Ψ ∈ GG is an LKG cover of G and u, v ∈ V (G) are such that uΨv, then at most one of u and v can be a Ψ-support to more than one Ψ-pendant vertex. After having come across graphs possessing an LKG cover, a natural and fundamental question that arises before us “Does every graph possess a least kernel graphoidal cover (an LKG cover)?” The answer to this question is “NO”. For this, we consider the graph G denoted by K44+ which is obtained by adjoining 4 new vertices of degree 1 at each vertex of complete graph K4 . We will prove in Theorem 2.5 that the graph K44+ does not possess an LKG cover i.e., whichever graphoidal cover Ψ of K44+ we may consider γΨ (K44+ ) = γiΨ (K44+ ).
Fig. 6. Example of a graph which does not possess an LKG cover.
Theorem 2.5 The graph G = K44+ does not possess any LKG cover. Proof. Let Ψ be any graphoidal cover of G. We claim that Ψ is not an LKG cover. Due to Lemma 2.4, it is enough to show that there exists a pair of Ψadjacent vertices in (G, Ψ) such that each of them is a Ψ-support to at least two Ψ-pendant vertices. Further, since every vertex in V (K4 ) is a support to 4 pendant vertices, therefore every vertex in V (K4 ) is Ψ-adjacent to at least two Ψ-pendant vertices (Lemma 2.2). Hence to prove that Ψ is not an LKG cover it is enough to prove that there exists a pair of Ψ-adjacent vertices in V (K4 ).
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Suppose no two vertices in V (K4 ) are Ψ-adjacent i.e., if P ∈ Ψ and E(P ) ∩ E(K4 ) = φ then either P is closed or at least one end vertex of P is a pendant vertex in G. Let V (K4 ) = {w, x, y, z} and Q be a Ψ-edge such that |E(R) ∩ E(K4 )| ≤ |E(Q) ∩ E(K4 )| for all R ∈ Ψ. Now we have two possibilities :: Case 1. Q is closed Then Q is a cycle of length 3. For if, Q is a cycle of length 4 then in that case at least one of the diagonals of the cycle Q must be a Ψ-edge, a contradiction to the assumption. Without loss of generality, let Q = (wxyw). Now let Qz be the Ψ-edge containing the edge yz. Clearly, Qz is not closed and hence at least one end vertex of Qz is a pendant vertex. Therefore z is an internal vertex of Qz . Since x is an internal vertex of Q, therefore xz must be a Ψ-edge, a contradiction. Case 2. Q is open. We claim that l(Q) ≤ 3. Suppose not, then l(Q) ≥ 4 and therefore at least three vertices say x, y, z of K4 are internal vertices of Q. Hence one of the edges xy, yz, zx must be a Ψ-edge, contradiction. Thus l(Q) ≤ 3. It further implies that |E(Q) ∩ E(K4 )| ≤ 2. Now we next claim that |E(Q) ∩ E(K4 )| = 2. For if |E(Q) ∩ E(K4 ) = 1, then any Ψ-edge can contain at most one edge of E(K4 ). Let Qw , Qy and Qz be the Ψ-edges containing the edges xw, xy and xz respectively. Since x can be an internal vertex to at most one Ψ-edge, with no loss in generality, we assume that x is an end vertex of Qy and Qz . But then y and z are internal vertices of Qy and Qz respectively and therefore yz must be a Ψ-edge, a contradiction to the assumption. Thus it follows that |E(Q) ∩ E(K4 )| = 2. Without loss of generality we let E(Q) ∩ E(K4 ) = {wx, xy}. Clearly, x is an internal vertex of Q. Also at least one of w and y is an internal vertex of Q. Without loss in generality, let y be an internal vertex of Q. Let Qz be the Ψ-edge containing the edge yz. Clearly, y is an end vertex of Qz and therefore z is an internal vertex of Qz . But then it implies that xz must be a Ψ-edge, a contradiction. Thus in both the cases we arrived at a contradiction. Hence our assumption is not correct and there exists a pair of Ψ-adjacent vertices in V (K4 ). Therefore it follows that Ψ cannot be an LKG cover. Since Ψ is an arbitrary graphoidal cover of G, therefore G does not possess any LKG cover. 2 Thus a graph may not possess a least kernel graphoidal cover. In case, the graph G possesses a least kernel graphoidal cover, we call G to be a least
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kernel graphoidal graph (or simply an LKG graph). After having seen that not every graph is an LKG graph, we move onto the next natural question. Do there exists graphs (classes) which always possess an LKG cover? Based on our earlier discussion, we deduce that every finite claw-free graph is an LKG graph and also that every γ(G)-definable infinite claw-free graph is also an LKG graph. Theorem 2.6 Every γ(G)-definable claw-free graph (finite or infinite) is an LKG graph. We further establish that every finite tree and every finite unicyclic graph possesses an LKG cover. To prove that every finite tree is an LKG graph, we will use spider decomposition of the tree to construct an LKG cover. We define a spider as a subdivision (or, a ‘homeomorph’) of a star K1,n for some positive integer n, where the center of K1,n is called the root, and the paths having the root as one of their ends and the other ends being the pendant vertices of the star being called legs of the so formed spider. It can be easily seen that every tree T can be decomposed into a set of spiders. Theorem 2.7 Every finite tree T admits an LKG cover. Proof. Let u0 be any vertex of T and let S(u0 ) denote a maximal spider with u0 as its root and the ends of its legs being the pendant vertices of T , ‘maximal’ in the sense that no other spider rooted at u0 can have more number of legs than S(u0 ). Let F1 be the spanning forest obtained by deleting all the edges of S(u0 ). If F1 does not have a vertex of degree at least one, then we are done. If not, choose any one of the vertices, say u1 , with degree at least one in F1 lying on any one of the legs of S(u0 ) and then let S(u1 ) be a maximal spider in F1 with u1 as its root. Delete all the edges of S(u1 ) from F1 to obtain another spanning forest F2 . Next, choose a vertex u2 either from a leg of S(u0 ) or from a leg of S(u1 ) with degree of u2 being at least one in F2 . We can iterate this process and obtain a sequence of maximal spiders S(u0 ), S(u1 ), . . . , S(ut ) in such a way that (i) the root ui of the ith spider S(ui ) lies on a leg of any one of the spiders previously formed; thus, the spanning forest Fi obtained at any stage i, 1 ≤ i ≤ t, is obtained by removing t the edges of the maximal spider S(ui−1 ) obtained at th the (i−1) stage, (ii) i=1 E(S(ui )) = E(T ), and (iii) E(S(ui ))∩E(S(uj )) = ∅ for all distinct indices i, j ∈ {1, 2, . . . , t}. Let Ψ consist of all the legs of all the spiders. Then, by the construction, Ψ is a graphoidal cover of G. Note that each of the Ψ-edges has one end vertex as a pendant vertex of T and the other end vertex is the root of the spider to which it belongs.
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Fig. 7. Graphoidally covered graph (T, Ψ) obtained by the spider decomposition of the tree T .
Let D consist of all the vertices of T except its pendant vertices. That is, D contains the roots of all the spiders constructed above as well as the vertices of (T, Ψ) with Ψ-degree zero. Then, clearly D is a Ψ-dominating set of (T, Ψ). We claim that D is a minimum Ψ-domset of (T, Ψ). Let D1 be a Ψ-domset, other than D and let x ∈ D \ D1 . That is, x ∈ V (T ) \ D1 . Since D1 is a Ψ-domset of (T, Ψ), x cannot be a vertex of Ψ-degree zero and hence must be the root of exactly one of the spiders constructed above. Also, since D1 is a Ψ-domset, there exists a vertex y ∈ D1 such that yΨx. This means, y is an end vertex of one of the legs of the spider rooted at x. But then this will imply that all the other ends of the spider rooted at x must also be in D1 . This implies |D1 | ≥ |D|, whence it follows that D is a minimum Ψ-domset of (T, Ψ), proving the claim. Further, since no two roots are Ψ-adjacent by the very construction, D is Ψ-independent as well. Thus, γΨ (T ) = γiΨ (T ), and the proof is complete. 2 In Fig. 7 we illustrate the technique used in the above proof to construct an LKG cover of the given tree. Note that D = {u0 , u1 , u2 , w0 , w1 , w2 , w3 } is a γΨ (T )(γiΨ (T ))-set of the graphoidally covered graph (T, Ψ), where Ψ is obtained by the spider decomposition of the tree T taking u0 as initial root. The technique of spider decomposition can be easily extended to show that every finite unicyclic graph possess an LKG cover. Theorem 2.8 Every finite unicyclic graph is an LKG graph. Proof. Let G be a unicyclic graph and Cn be the unique cycle of G. Let V (Cn ) = {v1 , ..., vn } be the vertex set of Cn and r(1 ≤ r ≤ n) be such that v1 , ..., vr has degree at least three and vr+1 , ..., vn has degree 2. For each (i = 1, 2, ..., r), let Ti be the non-trivial component of the graph G = G\E(Cn )
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containing vertex vi and Ψi be the graphoidal cover of Ti obtained by spider decomposition technique used in Theorem 2.7 taking vi as the initial root. Also, let Di be the γΨi (γiΨi )-set of (Ti , Ψi ) derived as in the proof of Theorem 2.7. Then Ψ = ∪ri=1 Ψi ∪ {Cn } is a graphoidal of G and the set D = ∪ri=1 Di ∪ {vr+1 , ..., vn } is a γΨ (G)-set (as well as γiΨ (G)-set) of (G, Ψ) which is Ψ-independent as well. Therefore the graphoidal cover Ψ of G is an LKG cover of G. Thus G is an LKG graph. 2
3
Concluding Remarks
In [1], Acharya and Gupta introduced the concept of domination to graphoidally covered graphs. In this paper, based on the fundamental problem of determining the graphs in which domination number equals the independent domination number, we introduced the concept of least kernel graphoidal graphs (LKG graphs) and thus extended the work done in [1]. Since this is the introduction of the concept, it provides researchers with lots of interesting problems to work on. We showed that every unicyclic graph is an LKG graph. Does it imply that every cactus is an LKG graph? Also, the example of K44+ implies that not every block-graph is an LKG graph, thus it becomes interesting to find out which block-graphs are LKG? Further, we observed that due to Allan-Laskar theorem every finite claw free happens to be an LKG graph with trivial graphoidal cover being an LKG cover. But then one may be tempted to ask is every graphoidal cover of a claw free an LKG cover? All these ideas evoke further interest to work on these concepts. We list some of the open problems based on all these discussions. Problem 3.1 Is every cactus an LKG graph? Problem 3.2 Characterize block-graphs which are LKG graphs. Problem 3.3 Is every graphoidal cover of a claw-free graph an LKG cover? Does there exist a claw-free graph G for which Ψ = E(G) ∈ GG is the only least kernel graphoidal cover of G?
References [1] B.D. Acharya and Purnima Gupta, Domination in graphoidal covers of a graph, Discrete Math., 206 (1999), 3–33.
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[2] B.D. Acharya and Purnima Gupta, Further results on domination in graphoidally covered graphs, AKCE Int. J. Graphs Combin., 4(2) (2007), 127– 138. [3] B.D. Acharya, E. Sampathkumar, Graphoidal covers and graphoidal covering number of a graph, Indian J. Pure Appl. Math., 18(10) (1987), 882–890. [4] B.D. Acharya, S. Arumugam and E. Sampathkumar, Graphoidal covers of a graph: A creative review. In: Graph Theory and Its Applications, (Eds.: S. Arumugam, B.D. Acharya and E. Sampathkumar), Tata McGraw-Hill Publ. Comp. Ltd., New Delhi, (1997), 1–28. [5] R.B Allan and Renu Laskar, On domination and independent domination numbers of a graph, Discrete Math., 23 (1978), 73–76. [6] S. Arumugam and Sithara Jerry, A note on independent domination in graphs, Bulletin of the Allahabad Mathematical Society, 23(1) (2008), 57–64. [7] S.T. Hedetniemi and R.C. Laskar, Topics in domination, Discrete Math., 48 (1990), North-Holland, 1990. [8] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc., New York, 1998. [9] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs: Advanced topics, Marcel Dekker, Inc., New York, 1998. [10] D.B. West, Introduction to Graph Theory, Prentice-Hall of India Private Limited, New Delhi, 1996.
Electronic Notes in Discrete Mathematics 53 (2016) 433–444 www.elsevier.com/locate/endm
Domination in Graphoidally Covered Graphs: Least-Kernel Graphoidal Covers Purnima Gupta 1 Department of Mathematics Sri Venkateswara College University of Delhi, Delhi, India
Rajesh Singh 3,2 Department Of Mathematics University Of Delhi, Delhi, India
(Dedicated to the sweet memories of Dr. B.D Acharya)
Abstract Given a graph G = (V, E) (not necessarily finite), a graphoidal cover of G means a collection Ψ of non-trivial paths in G called Ψ-edges, which are not necessarily open (not necessarily finite), such that every vertex of G is an internal vertex of at most one path in Ψ and every edge of G is in exactly one path in Ψ. The set of all graphoidal covers of a graph G = (V, E) is denoted by GG and for a given Ψ ∈ GG the ordered pair (G, Ψ) is called a graphoidally covered graph. Two vertices u and v of G are Ψ-adjacent if they are the ends of an open Ψ-edge. A set D of vertices in (G, Ψ) is Ψ-independent if no two vertices in D are Ψ-adjacent and is said to be Ψ-dominating if every vertex of G is either in D or is Ψ-adjacent to a vertex in D; G is γΨ (G)-definable (γiΨ (G)-definable) if (G, Ψ) has a finite Ψ-dominating (Ψ-independent Ψ-dominating) set. Clearly, if G is γiΨ (G)-definable, then G is γΨ (G)-definable and γΨ (G) ≤ γiΨ (G). Further if for a graphoidal cover Ψ of G, γΨ (G) = γiΨ (G) then we call Ψ as a least-kernel graphoidal cover of G (in short, http://dx.doi.org/10.1016/j.endm.2016.05.037 1571-0653/© 2016 Elsevier B.V. All rights reserved.
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an LKG cover of G). A natural question arises: “Does every graph possess an LKG cover?” We firstly exhibit that not every graph possesses an LKG cover and thereafter, in the quest to find graphs possessing an LKG cover, we proved that every finite tree and every finite unicyclic graph admits an LKG cover. We further extend Allan Laskar theorem to infinite graphs by showing every γ(G)-definable infinite claw free graph has an LKG cover. Keywords: Graphoidal Cover, Graphoidal Domination, Independent Domination.
1
Introduction
Throughout this paper we will consider simple graphs without loops as treated in most of the standard text-books on graph theory such as [10]. Further, unless mentioned otherwise, graphs will be assumed to be connected and infinite. In [3], Acharya and Sampathkumar introduced the notion of a graphoidal cover and graphoidal covering number of a finite graph and since then a lot of work has been done and hundreds of paper have come up in this short span. In 1999, Acharya and Gupta [1] extended the concept of graphoidal covers to infinite graphs and introduced the notion of domination to discrete structures, called graphoidally covered graphs. Given a graph G = (V, E) (not necessarily finite), a graphoidal cover [1] of G means a collection Ψ of non-trivial paths in G (i.e, having length at least one) called Ψ-edges, which are not necessarily open (not necessarily finite), such that the following conditions are satisfied: GC1 Every vertex of G is an internal vertex of at most one path in Ψ. GC2 Every edge of G is in exactly one path in Ψ The set of all graphoidal covers of a graph G is denoted by GG and for a given Ψ ∈ GG , the ordered pair (G, Ψ) is called a graphoidally covered graph. It may be noted that a Ψ-edge could possibly be infinite; in particular, it may be a one-way infinite path (or, the often so-called ‘ray’) or a two-way infinite path. Further, a finite open Ψ-edge has two distinct end vertices and 1
Email: [email protected] (corresponding author) Email: singh [email protected] 3 The author is thankful to Indian organization CSIR-UGC for providing the research grant to carry out the research. 2
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a closed Ψ-edge has only one end vertex, which is specified by Ψ. Next, a one-way infinite Ψ-edge has exactly one end vertex while a two-way infinite Ψ-edge has no end vertex. It may also be noted that there are three types of vertices in a graphoidally covered graph (G, Ψ), viz. (a) a vertex which is not an internal vertex of any Ψ-edge, called a Ψ-exterior vertex or a black vertex and shown as a small filled circle in the diagrammatic representation of (G, Ψ), (b) a vertex which is not an end-vertex of any Ψ-edge, called a Ψ-interior vertex or a white vertex and shown as a small unfilled circle in the diagrammatic representation of (G, Ψ), and (c) a vertex which is neither an exterior vertex nor an interior vertex of (G, Ψ), which means it is an internal vertex to a Ψ-edge as also an end-vertex to at least one other Ψ-edge, called a Ψ-composite vertex – in the diagrammatic representation, this vertex is represented as an unfilled circle with as many small tangents to the circle as the number of Ψ-edges for which this vertex is the end-vertex and each tangent indicates an end-vertex of the corresponding Ψ-edge. In Fig. 1 we give diagrammatic representation of four different graphoidal covers of K4 namely, E, Ψ1 , Ψ2 , Ψ3 where E is the edge set of G, Ψ1 = {bd, bca, badc}, Ψ2 = {bd, ac, badcb} and Ψ3 = {badc, acbd}. In Fig. 2 we have a graphoidal cover of an infinite tree in which every Ψ-edge is a one way infinite path.
Fig. 1. Different graphoidal covers of complete graph K4 .
Clearly, E ∈ GG and hence we call E =: E(G), the edge set of G, the trivial graphoidal cover of G; a graphoidal cover that is not trivial will be referred to as being nontrivial. Two distinct vertices u and v of G are Ψ-adjacent if they are the ends of an open Ψ-edge; a vertex is said to be self Ψ-adjacent if it is both the ‘initial’ and the ‘terminal’ end of a closed Ψ-edge (i.e., a cycle
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Fig. 2. An infinite tree, each of the Ψ-edges is a one way infinite path
in Ψ) – in either case, we represent the fact by writing uΨv. The Ψ-degree dΨ (u) of a vertex u is then defined as the number of open Ψ-edges having u as their end-vertex. A vertex is said to be a Ψ-pendant vertex if its Ψ-degree is one and a vertex is said to be a Ψ-support if it is Ψ-adjacent to at least one Ψ-pendant vertex. A set D of vertices in a graphoidally covered graph (G, Ψ) is a Ψ-dominating set (or ‘Ψ-domset’ for short) of G if every vertex of G is either in D or is Ψadjacent to a vertex in D; if (G, Ψ) contains a finite Ψ-dominating set, then the least cardinality of such a set is denoted as γΨ (G) and is called the Ψdomination number of (G, Ψ) ; such graphs G are called γΨ (G)-definable. A set D of vertices in a graphoidally covered graph (G, Ψ) is a Ψ-independent set if no pair of distinct vertices in D are Ψ-adjacent and a Ψ-dominating set D, which is also Ψ-independent is called a Ψ-kernel. If (G, Ψ) contains a finite Ψ-kernel then the least cardinality of a Ψ-kernel of (G, Ψ) is denoted by γiΨ (G) and then G is said to be ‘γiΨ (G)-definable’. In case (G, Ψ) is γΨ (G)- (γiΨ (G)) definable then it is often convenient to call a Ψ-dominating set (Ψ-kernel) consisting of γΨ (G) (γiΨ (G)) vertices a γΨ (G)-set (γiΨ (G)-set). Clearly, if (G, Ψ) is γiΨ (G)-definable then it is γΨ (G)-definable, but the converse may not be true for infinite graphs. In Fig. 3, we have an infinite graph G and a graphoidal cover Ψ of G such that G is γΨ (G)-definable but not γiΨ (G)definable. However, if G is an infinite claw free graph, then for the trivial graphoidal cover Ψ = E, γΨ (G)-definability implies γiΨ (G)-definability and in
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that case γΨ (G) = γiΨ (G). This fact can be proved using the technique used in the proof of Allan-Laskar Theorem [5] on claw free graphs.
Fig. 3. (G, Ψ), where Ψ = {uv} ∪ {ua2 a1 , ua3 a2 , ...} ∪ {vb2 b1 , vb3 b2 , ...}, is γΨ (G)-definable but not γiΨ (G)-definable and γΨ (G) = {u, v}.
For Ψ = E(G) we denote γE(G) (G) =: γ(G), and call it the domination number of G, and denote γiE(G) (G) =: γi (G), and call it the independent domination number of G, subject to the existence of these parameters, well in compatibility with these notations in the theory of domination in the case of finite graphs (e.g., see [8,9,7]).
2
Least Kernel Graphoidal Graphs
Given a graph G and a graphoidal cover Ψ of G such that G is γiΨ (G)-definable then obviously γΨ (G) ≤ γiΨ (G). (1) For the trivial graphoidal cover Ψ = E, the above inequality reduces to γE (G) ≤ γiE (G)
i.e.,
γ(G) ≤ γi (G).
We call a graphoidal cover Ψ ∈ GG to be a least kernel graphoidal cover (or simply an LKG cover ) of G if G is γiΨ (G)-definable and γΨ (G) = γiΨ (G).
(2)
For Ψ = E(G), this reduces to the well known equality γ(G) = γi (G), first considered by Allan-Laskar. Since by Allan Laskar Theorem [5], for every finite claw free graph γ(G) = γi (G), therefore the trivial graphoidal cover of every finite claw free graph is an LKG cover.
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In Fig. 4, we illustrate a graph G and a non-trivial graphoidal cover Ψ of G such that equality holds in (1) i.e., Ψ is an LKG cover of G. Since for the infinite graphoidally covered graph (G, Ψ) in Fig. 3, γΨ (G) = 2 and γiΨ (G) is not defined, therefore Ψ is not an LKG cover.
Fig. 4. {c, b1 , b2 } is a γψ1 (G)(γiψ1 (G)) set and ψ1 is an LKG Cover
In Fig. 5, we give an LKG cover of the infinite graph shown in Fig. 3. The set {u, v} forms a γΨ (G)-set as well as γiΨ (G)-set, where u is Ψ-adjacent to all ui ’s and v is Ψ-adjacent to all vi ’s.
Fig. 5. Ψ = {vuu0 v0 , vv2 v1 , vv3 v2 , ...} ∪ {uu1 u0 , uu2 u1 , ...} is an LKG cover of the infinite graph G.
Now before moving ahead, let us look at some of the basic but important observations that will prove handy while exploring least kernel graphoidal covers. We put them in a series of lemmas. Lemma 2.1 Every pendant vertex of a graph G is a Ψ-pendant vertex, for each Ψ ∈ GG .
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Lemma 2.2 If u ∈ V (G) is a support to n(≥ 3) pendant vertices, then for each Ψ ∈ GG , at least (n − 2) of these pendant vertices are Ψ-pendant vertices, with u as their common Ψ-support. Lemma 2.3 Let Ψ ∈ GG be such that G is γiΨ (G)-definable and if u ∈ V (G) is a Ψ-support to more than one Ψ-pendant vertices, then every γΨ (G)-set of G contains u. Lemma 2.4 If Ψ ∈ GG is an LKG cover of G and u, v ∈ V (G) are such that uΨv, then at most one of u and v can be a Ψ-support to more than one Ψ-pendant vertex. After having come across graphs possessing an LKG cover, a natural and fundamental question that arises before us “Does every graph possess a least kernel graphoidal cover (an LKG cover)?” The answer to this question is “NO”. For this, we consider the graph G denoted by K44+ which is obtained by adjoining 4 new vertices of degree 1 at each vertex of complete graph K4 . We will prove in Theorem 2.5 that the graph K44+ does not possess an LKG cover i.e., whichever graphoidal cover Ψ of K44+ we may consider γΨ (K44+ ) = γiΨ (K44+ ).
Fig. 6. Example of a graph which does not possess an LKG cover.
Theorem 2.5 The graph G = K44+ does not possess any LKG cover. Proof. Let Ψ be any graphoidal cover of G. We claim that Ψ is not an LKG cover. Due to Lemma 2.4, it is enough to show that there exists a pair of Ψadjacent vertices in (G, Ψ) such that each of them is a Ψ-support to at least two Ψ-pendant vertices. Further, since every vertex in V (K4 ) is a support to 4 pendant vertices, therefore every vertex in V (K4 ) is Ψ-adjacent to at least two Ψ-pendant vertices (Lemma 2.2). Hence to prove that Ψ is not an LKG cover it is enough to prove that there exists a pair of Ψ-adjacent vertices in V (K4 ).
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Suppose no two vertices in V (K4 ) are Ψ-adjacent i.e., if P ∈ Ψ and E(P ) ∩ E(K4 ) = φ then either P is closed or at least one end vertex of P is a pendant vertex in G. Let V (K4 ) = {w, x, y, z} and Q be a Ψ-edge such that |E(R) ∩ E(K4 )| ≤ |E(Q) ∩ E(K4 )| for all R ∈ Ψ. Now we have two possibilities :: Case 1. Q is closed Then Q is a cycle of length 3. For if, Q is a cycle of length 4 then in that case at least one of the diagonals of the cycle Q must be a Ψ-edge, a contradiction to the assumption. Without loss of generality, let Q = (wxyw). Now let Qz be the Ψ-edge containing the edge yz. Clearly, Qz is not closed and hence at least one end vertex of Qz is a pendant vertex. Therefore z is an internal vertex of Qz . Since x is an internal vertex of Q, therefore xz must be a Ψ-edge, a contradiction. Case 2. Q is open. We claim that l(Q) ≤ 3. Suppose not, then l(Q) ≥ 4 and therefore at least three vertices say x, y, z of K4 are internal vertices of Q. Hence one of the edges xy, yz, zx must be a Ψ-edge, contradiction. Thus l(Q) ≤ 3. It further implies that |E(Q) ∩ E(K4 )| ≤ 2. Now we next claim that |E(Q) ∩ E(K4 )| = 2. For if |E(Q) ∩ E(K4 ) = 1, then any Ψ-edge can contain at most one edge of E(K4 ). Let Qw , Qy and Qz be the Ψ-edges containing the edges xw, xy and xz respectively. Since x can be an internal vertex to at most one Ψ-edge, with no loss in generality, we assume that x is an end vertex of Qy and Qz . But then y and z are internal vertices of Qy and Qz respectively and therefore yz must be a Ψ-edge, a contradiction to the assumption. Thus it follows that |E(Q) ∩ E(K4 )| = 2. Without loss of generality we let E(Q) ∩ E(K4 ) = {wx, xy}. Clearly, x is an internal vertex of Q. Also at least one of w and y is an internal vertex of Q. Without loss in generality, let y be an internal vertex of Q. Let Qz be the Ψ-edge containing the edge yz. Clearly, y is an end vertex of Qz and therefore z is an internal vertex of Qz . But then it implies that xz must be a Ψ-edge, a contradiction. Thus in both the cases we arrived at a contradiction. Hence our assumption is not correct and there exists a pair of Ψ-adjacent vertices in V (K4 ). Therefore it follows that Ψ cannot be an LKG cover. Since Ψ is an arbitrary graphoidal cover of G, therefore G does not possess any LKG cover. 2 Thus a graph may not possess a least kernel graphoidal cover. In case, the graph G possesses a least kernel graphoidal cover, we call G to be a least
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kernel graphoidal graph (or simply an LKG graph). After having seen that not every graph is an LKG graph, we move onto the next natural question. Do there exists graphs (classes) which always possess an LKG cover? Based on our earlier discussion, we deduce that every finite claw-free graph is an LKG graph and also that every γ(G)-definable infinite claw-free graph is also an LKG graph. Theorem 2.6 Every γ(G)-definable claw-free graph (finite or infinite) is an LKG graph. We further establish that every finite tree and every finite unicyclic graph possesses an LKG cover. To prove that every finite tree is an LKG graph, we will use spider decomposition of the tree to construct an LKG cover. We define a spider as a subdivision (or, a ‘homeomorph’) of a star K1,n for some positive integer n, where the center of K1,n is called the root, and the paths having the root as one of their ends and the other ends being the pendant vertices of the star being called legs of the so formed spider. It can be easily seen that every tree T can be decomposed into a set of spiders. Theorem 2.7 Every finite tree T admits an LKG cover. Proof. Let u0 be any vertex of T and let S(u0 ) denote a maximal spider with u0 as its root and the ends of its legs being the pendant vertices of T , ‘maximal’ in the sense that no other spider rooted at u0 can have more number of legs than S(u0 ). Let F1 be the spanning forest obtained by deleting all the edges of S(u0 ). If F1 does not have a vertex of degree at least one, then we are done. If not, choose any one of the vertices, say u1 , with degree at least one in F1 lying on any one of the legs of S(u0 ) and then let S(u1 ) be a maximal spider in F1 with u1 as its root. Delete all the edges of S(u1 ) from F1 to obtain another spanning forest F2 . Next, choose a vertex u2 either from a leg of S(u0 ) or from a leg of S(u1 ) with degree of u2 being at least one in F2 . We can iterate this process and obtain a sequence of maximal spiders S(u0 ), S(u1 ), . . . , S(ut ) in such a way that (i) the root ui of the ith spider S(ui ) lies on a leg of any one of the spiders previously formed; thus, the spanning forest Fi obtained at any stage i, 1 ≤ i ≤ t, is obtained by removing t the edges of the maximal spider S(ui−1 ) obtained at th the (i−1) stage, (ii) i=1 E(S(ui )) = E(T ), and (iii) E(S(ui ))∩E(S(uj )) = ∅ for all distinct indices i, j ∈ {1, 2, . . . , t}. Let Ψ consist of all the legs of all the spiders. Then, by the construction, Ψ is a graphoidal cover of G. Note that each of the Ψ-edges has one end vertex as a pendant vertex of T and the other end vertex is the root of the spider to which it belongs.
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Fig. 7. Graphoidally covered graph (T, Ψ) obtained by the spider decomposition of the tree T .
Let D consist of all the vertices of T except its pendant vertices. That is, D contains the roots of all the spiders constructed above as well as the vertices of (T, Ψ) with Ψ-degree zero. Then, clearly D is a Ψ-dominating set of (T, Ψ). We claim that D is a minimum Ψ-domset of (T, Ψ). Let D1 be a Ψ-domset, other than D and let x ∈ D \ D1 . That is, x ∈ V (T ) \ D1 . Since D1 is a Ψ-domset of (T, Ψ), x cannot be a vertex of Ψ-degree zero and hence must be the root of exactly one of the spiders constructed above. Also, since D1 is a Ψ-domset, there exists a vertex y ∈ D1 such that yΨx. This means, y is an end vertex of one of the legs of the spider rooted at x. But then this will imply that all the other ends of the spider rooted at x must also be in D1 . This implies |D1 | ≥ |D|, whence it follows that D is a minimum Ψ-domset of (T, Ψ), proving the claim. Further, since no two roots are Ψ-adjacent by the very construction, D is Ψ-independent as well. Thus, γΨ (T ) = γiΨ (T ), and the proof is complete. 2 In Fig. 7 we illustrate the technique used in the above proof to construct an LKG cover of the given tree. Note that D = {u0 , u1 , u2 , w0 , w1 , w2 , w3 } is a γΨ (T )(γiΨ (T ))-set of the graphoidally covered graph (T, Ψ), where Ψ is obtained by the spider decomposition of the tree T taking u0 as initial root. The technique of spider decomposition can be easily extended to show that every finite unicyclic graph possess an LKG cover. Theorem 2.8 Every finite unicyclic graph is an LKG graph. Proof. Let G be a unicyclic graph and Cn be the unique cycle of G. Let V (Cn ) = {v1 , ..., vn } be the vertex set of Cn and r(1 ≤ r ≤ n) be such that v1 , ..., vr has degree at least three and vr+1 , ..., vn has degree 2. For each (i = 1, 2, ..., r), let Ti be the non-trivial component of the graph G = G\E(Cn )
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containing vertex vi and Ψi be the graphoidal cover of Ti obtained by spider decomposition technique used in Theorem 2.7 taking vi as the initial root. Also, let Di be the γΨi (γiΨi )-set of (Ti , Ψi ) derived as in the proof of Theorem 2.7. Then Ψ = ∪ri=1 Ψi ∪ {Cn } is a graphoidal of G and the set D = ∪ri=1 Di ∪ {vr+1 , ..., vn } is a γΨ (G)-set (as well as γiΨ (G)-set) of (G, Ψ) which is Ψ-independent as well. Therefore the graphoidal cover Ψ of G is an LKG cover of G. Thus G is an LKG graph. 2
3
Concluding Remarks
In [1], Acharya and Gupta introduced the concept of domination to graphoidally covered graphs. In this paper, based on the fundamental problem of determining the graphs in which domination number equals the independent domination number, we introduced the concept of least kernel graphoidal graphs (LKG graphs) and thus extended the work done in [1]. Since this is the introduction of the concept, it provides researchers with lots of interesting problems to work on. We showed that every unicyclic graph is an LKG graph. Does it imply that every cactus is an LKG graph? Also, the example of K44+ implies that not every block-graph is an LKG graph, thus it becomes interesting to find out which block-graphs are LKG? Further, we observed that due to Allan-Laskar theorem every finite claw free happens to be an LKG graph with trivial graphoidal cover being an LKG cover. But then one may be tempted to ask is every graphoidal cover of a claw free an LKG cover? All these ideas evoke further interest to work on these concepts. We list some of the open problems based on all these discussions. Problem 3.1 Is every cactus an LKG graph? Problem 3.2 Characterize block-graphs which are LKG graphs. Problem 3.3 Is every graphoidal cover of a claw-free graph an LKG cover? Does there exist a claw-free graph G for which Ψ = E(G) ∈ GG is the only least kernel graphoidal cover of G?
References [1] B.D. Acharya and Purnima Gupta, Domination in graphoidal covers of a graph, Discrete Math., 206 (1999), 3–33.
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[2] B.D. Acharya and Purnima Gupta, Further results on domination in graphoidally covered graphs, AKCE Int. J. Graphs Combin., 4(2) (2007), 127– 138. [3] B.D. Acharya, E. Sampathkumar, Graphoidal covers and graphoidal covering number of a graph, Indian J. Pure Appl. Math., 18(10) (1987), 882–890. [4] B.D. Acharya, S. Arumugam and E. Sampathkumar, Graphoidal covers of a graph: A creative review. In: Graph Theory and Its Applications, (Eds.: S. Arumugam, B.D. Acharya and E. Sampathkumar), Tata McGraw-Hill Publ. Comp. Ltd., New Delhi, (1997), 1–28. [5] R.B Allan and Renu Laskar, On domination and independent domination numbers of a graph, Discrete Math., 23 (1978), 73–76. [6] S. Arumugam and Sithara Jerry, A note on independent domination in graphs, Bulletin of the Allahabad Mathematical Society, 23(1) (2008), 57–64. [7] S.T. Hedetniemi and R.C. Laskar, Topics in domination, Discrete Math., 48 (1990), North-Holland, 1990. [8] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc., New York, 1998. [9] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs: Advanced topics, Marcel Dekker, Inc., New York, 1998. [10] D.B. West, Introduction to Graph Theory, Prentice-Hall of India Private Limited, New Delhi, 1996.