Traceability of connected domination critical graphs
Introduction
A dominating set in a graph G is a set S of vertices of G such that every vertex in V(G)∖S is adjacent to at least one vertex in S. The domination number of G, denoted by γ(G), is the minimum cardinality of a dominating set of G. A graph G is said to be k-γ-critical if and for every pair of non-adjacent vertices u and v of G. Such a graph G is called a domination critical graph. If S is a dominating set of G, we write S≻G, and if we also write v≻G rather than {v}≻G. The concept of domination and its variations have been widely studied in the literature; a rough estimate says that it occurs in more than 6000 papers to date. A thorough treatment of the fundamentals of domination theory in graphs can be found in the books [12], [13].
A connected dominating set, abbreviated a CD-set, of a connected graph G is a dominating set S of G such that the subgraph G[S] induced by S is connected. The connected domination number of G, denoted by γc(G), is the minimum cardinality of a CD-set of G. A CD-set of G of cardinality γc(G) is called a γc-set of G. A graph G is said to be k-γc-critical if and for every pair of non-adjacent vertices u and v of G. Such a graph G is called a connected domination critical graph. If S is a CD-set of G, we write S≻cG, and if we also write v≻cG rather than {v}≻cG. The concept of connected domination was studied at least in the early 1970s, although it was first formally defined by Sampathkumar and Walikar in their 1979 paper [23]. Subsequently over the past forty years, the connected domination number has been extensively studied in the literature; a rough estimate says that it occurs in more than 400 papers to date. For a small sample of papers on the connected domination we refer the reader to [3], [7], [8], [21], [22], [24].
We remark that the concept of connected domination in graphs is application driven, as evidenced by the earlier papers on the concept. For example, Wu and Li [28] showed that connected dominating sets are useful in the computation of routing for mobile ad hoc networks. In this application, a minimum connected dominating set is used as a backbone for communications, and vertices that are not in this set communicate by passing messages through neighbors that are in the set.
We also remark that finding connected dominating sets and Steiner trees in a graph are closely related [5], [6]. Moreover, determining the connected domination number of a connected graph G is equivalent to finding the largest possible number of leaves among all spanning trees of G. A maximum leaf spanning tree of G is a spanning tree that has the largest possible number of leaves among all spanning trees of G, and the max leaf number, denoted by ℓmax(G), of G is the number of leaves in a maximum leaf spanning tree of G. Since the problems of a connected dominating set and a maximum leaf spanning tree are closely connected. The maximum leaf spanning tree problem is MAX-SNP hard, implying that no polynomial time approximation scheme is likely [11]. We remark, however, that both the minimum connected dominating set problem and the maximum leaf spanning tree problem are fixed-parameter tractable [2]. The connected dominating set problem is polynomially solvable for distance-hereditary graphs [6].
For notation and graph theory terminology, we in general follow [14]. Specifically, let be a graph with vertex set and edge set and let v be a vertex in V. A neighbor of a vertex is a vertex adjacent to it. The open neighborhood of v is the set NG(v) of all neighbors of v, and so and the closed neighborhood of v is . A vertex v is said to dominate a vertex u in G if or if u is a neighbor of v. The degree of a vertex v is |NG(v)| and is denoted by dG(v). An end vertex is a vertex of degree 1 and a support vertex is a vertex adjacent to an end vertex. For a set S of vertices in G, the subgraph induced by S in G is denoted by G[S]. If G is a graph, the complement of G, denoted by is formed by taking the vertex set of G and joining two vertices by an edge whenever they are not joined in G. If the graph G is clear from the context, we omit it in the above expressions. For example, we write N(v) and N[v] rather than NG(v) and NG[v], respectively. We use the standard notation .
Two vertices u and v in a graph G are connected if there exists a (u, v)-path in G. A graph G is connected if every two vertices in G are connected. We denote the number of components in a graph G by ω(G). The distance dG(u, v) between two vertices u and v in a connected graph G is the length of a shortest (u, v)-path in G. A hamiltonian cycle (respectively, hamiltonian path) of a graph is a cycle (path) passing through all vertices of the graph. A graph G is traceable if it contains a hamiltonian path. Moreover, a graph G is hamiltonian if it contains a hamiltonian cycle. For any subgraph F of G and distinct vertices a and b of G, aPFb denotes an (a, b)-path in G all of whose internal vertices are in V(F). We note that a and b need not be in V(F). If P is an (a, b)-path in G, we sometimes write the path P by aPb to indicate the start and end vertices of the path P.
We denote the path, cycle, and complete graph on n vertices by Pn, Cn, and Kn, respectively, and we denote the complete bipartite graph with partite sets of cardinality n and m by Kn,m. A star is the graph K1,k, where k ≥ 1. The graph K1,3 is called a claw. A graph G is claw-free if it does not contain a claw as an induced subgraph. A tree is a connected graph with no cycle.
For vertex subsets X, Y⊆V(G), we let NY(X) be the set of all vertices in Y that have a neighbor that belongs to X in G, that is, for some x ∈ X}. For a subgraph H of G, we use NY(H) instead of NY(V(H)) and we use NH(X) instead of NV(H)(X). If we use NY(x) instead of NY({x}). The open neighborhood of a set S of vertices in G is the set and its closed neighborhood is the set .
A subset S⊆V(G) is a vertex cut set of G if the number of components of is more than the number of components of G; that is, . In particular, if then v is called a cut-vertex of G. We let ζ(G) be the number of cut-vertices of G. When no ambiguity can occur, we write ζ instead of ζ(G). A block of a graph G is a maximal connected subgraph of G has no cut-vertex of its own. Thus, a block is a maximal 2-connected subgraph of G. Any two blocks of a graph have at most one vertex in common, namely a cut-vertex. A block of G containing exactly one cut-vertex of G is called an end block. If a connected graph contains a single block, we call the graph itself a block.
For ℓ ≥ 2 and a finite sequence of vertex disjoint graphs, we let the join G1∨⋅⋅⋅∨Gℓ be the graph obtained from the disjoint union of by joining each vertex in Gi to all vertices in for . If then we write . Moreover, for vertex disjoint graphs G1 and G2 and for a subgraph H of G2, the join G1∨HG2 is the graph obtained from the disjoint union of G1 and G2 by joining each vertex in G1 to each vertex in H.
A study of properties of domination critical graphs was initiated by Sumner and Blitch in their classical 1983 paper [25]. Among other results, they showed that every connected -critical graph of even order contains a perfect matching. Wojcicka [27] subsequently studied hamiltonian properties of domination critical graphs and showed every connected -critical graph on at least seven vertices is traceable. Favaron et al. [9], Flandrin et al. [10] and Tian et al. [26] proved further that all connected -critical graphs with minimum degree at least 2 are hamiltonian. Motivated in part by these results, Sumner and Wojcicka (Chapter 16 in [12]) conjectured in 1998 that all -connected k-γ-critical graphs are hamiltonian for all k ≥ 4. However, their conjecture was disproved seven years later by Yuansheng et al. [29] who constructed a 3-connected -critical non-hamiltonian graph containing 13 vertices. On the positive side, Kaemawichanurat and Caccetta [19] proved the Sumner-Wojcicka Conjecture is true if and the graphs are claw-free.
Kaemawichanurat [15] initiated a study of connected domination critical graphs. Hamiltonian properties of connected domination critical graphs were subsequently studied by Kaemawichanurat, Caccetta and Ananchuen [20] who showed that every 2-connected k-γc-critical graph is hamiltonian for all k ∈ [3]. Further, they constructed k-γc-critical graphs that are non-hamiltonian for all k ≥ 4. Recently, Kaemawichanurat and Caccetta [19] proved that every 2-connected -critical claw-free graph is hamiltonian, and they constructed 2-connected k-γc-critical claw-free graphs that are non-hamiltonian for all k ≥ 5. For 5 ≤ k ≤ 6, they proved that every 3-connected k-γc-critical claw-free graph is hamiltonian. Recall that ζ(G) denotes the number of cut-vertices of G, and that if the graph G is clear from the context, we simply write ζ instead of ζ(G). Kaemawichanurat and Ananchuen [18] showed that a connected domination critical graph cannot have too many cut-vertices. Theorem 1 ([18]) For k ≥ 2, every k-γc-critical graph has at most cut-vertices, that is, .
Section snippets
Main result
Our aim in this paper is to determine a connection between the traceability of a k-γc-critical graph and the number of cut-vertices in the graph. More precisely, we shall prove the following result. Theorem 2 For k ≥ 4 and every k-γc-critical graph with ζ cut-vertices has a hamiltonian path if and only if .
Preliminary results
In this section, we present some preliminary results that we will need to prove our main theorem, namely Theorem 2. The following result is a simple exercise in most graph theory textbooks. Observation 1 Let G be a graph and let S be a nonempty proper subset of V(G). If G is traceable, then .
By Observation 1, if S is a vertex cut set of a graph G satisfying then G is non-traceable. Kaemawichanurat, Caccetta and Ananchuen [20] showed that connected domination critical graphs with
Traceability of k-γc-critical graphs
In this section, we show that, for k ≥ 4 and every k-γc-critical graph with ζ cut-vertices contains a hamiltonian path. We first prove basic properties of k-γc-critical graphs.
In what follows, let B be a graph in the class of order n0 and let the vertex b be the head of B. For notational convenience, we sometimes rename the vertex b as the vertex . We show first that there exists a hamiltonian path in B that contains the vertex b as one of its ends. Lemma 3 If with the vertex b as
k-γc-Critical graphs which are non-traceable
In this section, we establish the realizability result that for k ≥ 4, there exist k-γc-critical graphs which are non-traceable containing ζ vertices for all . For this purpose, for k ≥ 3 we introduce a class of k-γc-critical graphs such that, for every graph and every integer ℓ ≥ 1, there exists a -critical graph that contains G as an induced subgraph. Further, we construct a class of graphs for all s ≥ 6.
The class for k ≥ 3. A k-γc-critical graph G is in
References (29)
- et al.
Connected domination critical graphs
Appl. Math. Letters
(2004) - et al.
Permutation graphs: connected domination and steiner trees
Discrete Math.
(1990) - et al.
Bounds on the connected domination number of a graph
Discrete Appl. Math.
(2013) - et al.
Connected domination of regular graphs
Discrete Math.
(2009) - et al.
Some properties of 3-domination-critical graphs
Discrete Math.
(1999) - et al.
A short note on the approximability of the maximum leaves spanning tree problem
Inf. Process. Lett.
(1994) - et al.
Hamiltonicity in 3-domination critical graphs with
Discrete Appl. Math.
(1999) - et al.
Hamiltonian properties of domination critical graphs
Theory
(1990) - et al.
On calculating connected dominating set for efficient routing in ad hoc wireless networks
Proceedings of the 3rd International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications
(1999) On domination critical graphs with cut vertices having connected domination number 3
Int. Math. Forum
(2007)
A parameterized measure-and-conquer analysis for finding a k-leaf spanning tree in an undirected graph
Discrete Math. Theor. Comput. Sci.
Connected domination and spanning trees with many leaves
SIAM J. Discrete Math.
Distance-hereditary graphs, steiner trees, and connected domination
SIAM J. Comput.
Independence and hamiltonicity in 3-domination-critical graphs
Theory
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This paper is dedicated to Professor Louis Caccetta on the occasion of his 70th birthday.
- 1
Research supported in part by the University of Johannesburg.
- 2
Research supported by Thailand Research Fund (MRG 6280223)