Abstract
A paired-dominating set of a graph G is a dominating set D with the additional requirement that the induced subgraph G[D] contains a perfect matching. We prove that the vertex set of every claw-free cubic graph can be partitioned into two paired-dominating sets.
Highlights
Except when otherwise specified, we shall deal with simple graphs G = (V, E) with vertex set V and edge set E
Definition 1 A set D ⊆ V is a dominating set if every vertex outside D has at least one neighbor in D
A dominating set is paired-dominating if it induces a subgraph having a perfect matching
Summary
A dominating set D is a total dominating set if it induces an isolate-free subgraph. If S is a maximal stable (independent) set in an isolate-free graph G, S and V \ S are two disjoint dominating sets. Theorem 1 If G is a claw-free cubic graph, its vertex set can be partitioned into two paired-dominating sets. Definition 3 For arbitrary graphs G and L, G is L-free if it does not contain any induced subgraph isomorphic to L. The claw is a four-vertex three-edge graph with a center and three vertices adjacent to it. A pair of jewels is a graph containing two vertex-disjoint induced subgraphs which are one of the diamond and the brill, supplemented by two edges between them so that we obtain a cubic graph. An edge is monochromatic if its endpoints have the same color
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