Mutual transferability for [formula omitted]-domination on strongly chordal graphs and cactus graphs

Abstract

This paper studies a variation of domination in graphs called (F,B,R)-domination. Let G=(V,E) be a graph and V be the disjoint union of F, B, and R, where F consists of free vertices, B consists of bound vertices, and R consists of required vertices. An (F,B,R)-dominating set of G is a subset D⊆V such that R⊆D and each vertex in B−D is adjacent to some vertex in D. An (F,B,R)-2-stable set of G is a subset S⊆B such that S∩N(R)=0̸ and every two distinct vertices x and y in S have distance d(x,y)>2. We prove that if G is strongly chordal, then αF,B,R,2(G)=γF,B,R(G)−|R|, where γF,B,R(G) is the minimum cardinality of an (F,B,R)-dominating set of G and αF,B,R,2(G) is the maximum cardinality of an (F,B,R)-2-stable set of G. Let D1→∗D2 denote D1 being transferable to D2. We prove that if G is a connected strongly chordal graph in which D1 and D2 are two (F,B,R)-dominating sets with |D1|=|D2|, then D1→∗D2. We also prove that if G is a cactus graph in which D1 and D2 are two (F,B,R)-dominating sets with |D1|=|D2|, then D1∪{1⋅extra}→∗D2∪{1⋅extra}, where ∪{1⋅extra} means adding one extra element.