Domination Parameters of Graph Covers

TL;DR

The paper studies how domination parameters change when G is a $k$-fold cover of F, formalizing lifts of subgraphs and key degree equalities. It proves the general upper bound $\gamma(G) \le k\,\gamma(F)$ with tightness in regular graphs and a greedy-based lower bound $\gamma(G) \ge \frac{k\,\gamma(F)}{H(\Delta)}$, and extends the same analysis to $\gamma_t$ and $\gamma_c$. It also yields a bound for $\gamma_t$ and $\gamma_c$: $\gamma_t(G) \le k\,\gamma_t(F)$ and $\gamma_c(G) \le k(\gamma_c(F)+2) - 2$, plus related lower bounds and counterexamples. The authors propose a conjecture that a universal constant $c>0$ exists with $\gamma(G) \ge c\,k\,\gamma(F)$ for all $k$-fold covers, with $c = 3/5$ suggested by cycles and regular graphs, and discuss implications for domination in graph covers.

Abstract

A graph $G$ is a \emph{cover} of a graph $F$ if there exists an onto mapping $π: V(G) \to V(F)$, called a (\emph{covering}) \emph{projection}, such that $π$ maps the neighbours of any vertex $v$ in $G$ bijectively onto the neighbours of $π(v)$ in $F$. This paper is the first attempt to study the connection between domination parameters and graph covers. We focus on the domination number, the total domination number, and the connected domination number. We prove upper and lower bounds for the domination parameters of $G$. Moreover, we propose a conjecture on the lower bound for the domination number of $G$ and provide evidence to support the conjecture.

Figures (3)

• Figure 1: Dodecahedron (left) is a (double) cover of the Petersen graph (right).
• Figure 2: The graph $N$.
• Figure 3: The graph $H$.

Theorems & Definitions (28)

• Proposition 1
• Theorem 2: walikar1979recent
• Theorem 3: reed1996paths
• Theorem 4: bujtas2019domination
• Theorem 5: cockayne1980totalwalikar1979recent
• Theorem 6: archdeacon2004somechvatal1992smalltuza1990covering
• Theorem 7: eustis2016independence
• Theorem 8: vazirani2002approximation
• Theorem 9
• proof