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A subset S of vertices of a graph G with no isolated vertex is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex in V (G) − S is also adjacent to a vertex in V (G) − S. The total restrained domination number of G is the minimum cardinality of a total restrained dominating set of G. In this paper we initiate the… (More)
A topological index of graph G is a numerical parameter related to G which characterizes its molecular topology and is usually graph invariant. In the field of quantitative structure-activity (QSAR)/quantitative structure-activity structure-property (QSPR) research, theoretical properties of the chemical compounds and their molecular topological indices… (More)
Abstract. For integers p, q, s with p ≥ q ≥ 2 and s ≥ 0, let K−s 2 (p, q) denote the set of 2−connected bipartite graphs which can be obtained from Kp,q by deleting a set of s edges. In this paper, we prove that for any graph G ∈ K−s 2 (p, q) with p ≥ q ≥ 3 and 1 ≤ s ≤ q−1, if the number of 3-independent partitions of G is 2p−1 + 2q−1 + s + 4, then G is… (More)
A graph G is said to be (γ, k)-critical if γ(G−S) < γ(G) for any set S of k vertices and domination number γ. Properties of (γ, k)-critical graphs are studied for k ≥ 3. Ways of constructing a (γ, k)-critical graph from smaller (γ, k)-critical graphs are presented.
A subset S of vertices in a graph G is a global total dominating set, or just GTDS, if S is a total dominating set of both G and G. The global total domination number γgt(G) of G is the minimum cardinality of a GTDS of G. In this paper, we show that the decision problem for γgt(G) is NP-complete, and then characterize graphs G of order n with γgt(G) = n− 1.
A K4-homeomorph is a subdivision of the complete graph with four vertices (K4). Such a homeomorph is denoted by K4(a,b,c,d,e,f) if the six edges of K4 are replaced by the six paths of length a,b,c,d,e,f, respectively. In this paper, we discuss the chromaticity of a family of K4-homeomorphs with girth 10. We also give sufficient and necessary condition for… (More)
In this paper, we discuss a pair of chromatically equivalent of K4-homeomorphs of girth 11, that is, K4(1, 3, 7, d, e, f) and K4(1, 3, 7, d′, e′, f ′). As a result, we obtain two infinite chromatically equivalent non-isomorphic K4-homeomorphs. Mathematical Subject Classification: 05C15