# Mutual transferability for (F,B,R)-domination on strongly chordal graphs and cactus graphs

Kuan Ting Chu, Wu-Hsiung Lin*, Chiuyuan Chen

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

## 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 α F,B,R,2 (G)=γ F,B,R (G)−|R|, where γ F,B,R (G) is the minimum cardinality of an (F,B,R)-dominating set of G and α F,B,R,2 (G) is the maximum cardinality of an (F,B,R)-2-stable set of G. Let D 1 →∗D 2 denote D 1 being transferable to D 2 . We prove that if G is a connected strongly chordal graph in which D 1 and D 2 are two (F,B,R)-dominating sets with |D 1 |=|D 2 |, then D 1 →∗D 2 . We also prove that if G is a cactus graph in which D 1 and D 2 are two (F,B,R)-dominating sets with |D 1 |=|D 2 |, then D 1 ∪{1⋅extra}→∗D 2 ∪{1⋅extra}, where ∪{1⋅extra} means adding one extra element.

Original language English 41-52 12 Discrete Applied Mathematics 259 https://doi.org/10.1016/j.dam.2018.12.034 Published - 30 Apr 2019

## Keywords

• Cactus graphs
• Domination
• Stability
• Strongly chordal graphs
• Transferability