On the Relation Between the Domination Number and Edge Domination Number of Trees and Claw-Free Cubic Graphs - Zhuo Pan; Peng Pan; Chongshan Tie (Mathematics)
For a connected graph <semantics>G=(V,E)</semantics>, the dominating set in graph G is a subset of vertices <semantics>F⊂V</semantics> such that every vertex of <semantics>V−F</semantics> is adjacent to at least one vertex of F. The minimum cardinality of a dominating set of G, denoted by <semantics>γ(G)</semantics>, is the domination number of G. The edge dominating set in graph G is a subset of edges <semantics>S⊂E</semantics> such that every edge of <semantics>E−S</semantics> is adjacent to at least one edge of S. The minimum cardinality of an edge dominating set of G, denoted by <semantics>γ′(G)</semantics>, is the edge domination number of G. In this paper, we characterize all trees and claw-free cubic graphs with equal domination and edge domination numbers, respectively.