On independent domination of regular graphs

Abstract

The domination number of a graph G $G$, denoted γ ( G ) $\gamma (G)$, is the minimum size of a dominating set of G $G$, and the independent domination number of G $G$, denoted i ( G ) $i(G)$, is the minimum size of a dominating set of G $G$ that is also independent. Let k ≥ 4 $k\ge 4$ be an integer. Generalizing a result on cubic graphs by Lam, Shiu, and Sun, we prove that i ( G ) ≤ k − 1 2 k − 1 false| V ( G ) false| $i(G)\le \frac{k-1}{2k-1}|V(G)|$ for a connected k $k$‐regular graph G $G$ that is not K k , k ${K}_{k,k}$, which is tight for k = 4 $k=4$. This answers a question by Goddard et al. in the affirmative. We also show that i ( G ) γ ( G ) ≤ k 3 − 3 k 2 + 2 2 k 2 − 6 k + 2 $\frac{i(G)}{\gamma (G)}\le \frac{{k}^{3}-3{k}^{2}+2}{2{k}^{2}-6k+2}$ for a connected k $k$‐regular graph G $G$ that is not K k , k ${K}_{k,k}$, strengthening upon a result of Knor, Škrekovski, and Tepeh. In addition, we prove that a graph G $G$ with maximum degree at most 4 satisfies i ( G ) ≤ 5 9 false| V ( G ) false| $i(G)\le \frac{5}{9}|V(G)|$, which is also tight.

A tight bound for independent domination of cubic graphs without 4‐cycles

“…The next natural step is to try to improve the upper bound on the independent domination number when the balanced complete bipartite graph is excluded. Extending a result by Lam, Shiu, and Sun [12], Cho, Choi, and Park [4] recently proved the following result: Theorem 1.1 (Cho et al [4]). For k 3…”

The Lower Bipartite Number of a Graph

“…For example: THEOREM 4. If is a cubic triangle-free graph of order , then LB( ) ≥ 3 5 , and this is best possible. Proof.…”

“…In contrast, for bipartite domination it is easy to show that bip ( ) = ( ) (see [1]) for any cubic graph. Questions for independent domination are investigated in [3].…”

“…Maybe its lower bipartite number is 1/3 the order-this value is achieved by placing kings everywhere on row 2, 5, 8, etc. In another direction, there are multiple open questions about independent domination number of -regular graphs for small (see for example [3,6]); such questions could also be considered for bipartite domination.…”

Independent domination of graphs with bounded maximum degree

“…Obviously, i(G) ≥ γ(G) for every graph G, and there has been a number of results comparing i(G) to γ(G). See, for example, [2,3].…”