Efficient (j,k)-Domination in Regular Graphs

TL;DR

This paper investigates the conditions for the existence of efficient (j,k)-dominating functions in regular graphs, especially focusing on Hamming graphs over prime fields, extending previous results on Cayley graphs.

Contribution

It establishes that for prime q, the necessary divisibility condition for efficient (1,k)-domination in regular graphs is also sufficient in Hamming graphs, extending prior work on Cayley graphs.

Findings

For prime q, the divisibility condition is sufficient for H(q,d).

The result generalizes Lee's work on Cayley graphs.

Difficulties are discussed for prime power q, not prime.

Abstract

Rubalcaba and Slater (Robert R. Rubalcaba and Peter J. Slater. Efficient (j,k)-domination. Discuss. Math. Graph Theory, 27(3):409-423, 2007.) define a $(j,k)$-dominating function on graph $X$ as a function $f:V(X)\rightarrow \{0,\ldots,j\}$ so that for each $v\in V(X)$, $f(N[v])\geq k$, where $N[v]$ is the closed neighbourhood of $v$. Such a function is efficient if all of the vertex inequalities are met with equality. They give a simple necessary condition for efficient domination, namely: if $X$ is an $r$-regular graph on $n$ vertices that has an efficient $(1,k)$-dominating function, then the size of the corresponding dominating set divides $n\cdot k$. The Hamming graph $H(q,d)$ is the graph on the vectors $\mathbb{Z}_q^d$ where two vectors are adjacent if and only if they are at Hamming distance $1$. We show that if $q$ is prime, then the previous necessary condition is sufficient for $H(q,d)$ to have an efficient $(1,k)$-dominating function. This result extends a result of Lee (Jaeun Lee. Independent perfect domination sets in Cayley graphs. J. Graph Theory, 37(4):213-219, 2001.) on independent perfect domination in Cayley graphs. We mention difficulties that arise for $H(q,d)$ when $q$ is a prime power but not prime.