Simultaneous Domination in Graphs

TL;DR

This paper studies the simultaneous domination number in multiple graphs sharing the same vertices, providing bounds and analyzing special cases such as regular graphs, unions of complete graphs, and cycles.

Contribution

It introduces general upper bounds for the simultaneous domination number and explores specific cases including regular graphs, unions of complete graphs, and cycles.

Findings

Established upper bounds on the simultaneous domination number.

Analyzed cases where factors are r-regular graphs or unions of K_r.

Investigated the case when each factor is a cycle.

Abstract

Let $F_1, F_2, ..., F_k$ be graphs with the same vertex set $V$. A subset $S \subseteq V$ is a simultaneous dominating set if for every $i$, $1 \le i \le k$, every vertex of $F_i$ not in $S$ is adjacent to a vertex in $S$ in $F_i$; that is, the set $S$ is simultaneously a dominating set in each graph $F_i$. The cardinality of a smallest such set is the simultaneous domination number. We present general upper bounds on the simultaneous domination number. We investigate bounds in special cases, including the cases when the factors, $F_i$, are $r$-regular or the disjoint union of copies of $K_r$. Further we study the case when each factor is a cycle.