Domination in Functigraphs

TL;DR

This paper investigates domination numbers in functigraphs, a graph class generalizing permutation graphs, exploring bounds, specific functions achieving these bounds, and detailed analysis for cycles.

Contribution

It characterizes domination number bounds in functigraphs and identifies functions that attain these bounds, especially for cycles.

Findings

Domination number of functigraphs ranges between γ(G) and 2γ(G).

Certain functions achieve the lower and upper bounds of domination numbers.

Detailed analysis provided for cycles in functigraphs.

Abstract

Let $G_1$ and $G_2$ be disjoint copies of a graph $G$, and let $f: V(G_1) \rightarrow V(G_2)$ be a function. Then a \emph{functigraph} $C(G, f)=(V, E)$ has the vertex set $V=V(G_1) \cup V(G_2)$ and the edge set $E=E(G_1) \cup E(G_2) \cup \{uv \mid u \in V(G_1), v \in V(G_2), v=f(u)\}$. A functigraph is a generalization of a \emph{permutation graph} (also known as a \emph{generalized prism}) in the sense of Chartrand and Harary. In this paper, we study domination in functigraphs. Let $\gamma(G)$ denote the domination number of $G$. It is readily seen that $\gamma(G) \le \gamma(C(G,f)) \le 2 \gamma(G)$. We investigate for graphs generally, and for cycles in great detail, the functions which achieve the upper and lower bounds, as well as the realization of the intermediate values.