On the Independent Domination Number of Regular Graphs

Abstract

A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. In this paper, we consider questions about independent domination in regular graphs.

Independent Domination in Cubic Graphs

“…We note that by Claim 12 we have {v 3 , v 3 } ∩ {v 4 , v 4 } = ∅ and by Claims 8 and 10 we have {v 3 3 ∈ E(G), then wv 4 ∈ E(G) produces a 4-cycle in G and wv 4 ∈ E(G) produces a 6-cycle in G, contradicting Claims 10 and 11 respectively. Therefore, wv 3 / ∈ E(G) and, analogously, …”

“…The graphs K 2,3 and C 5 K 2 are depicted in Figure 2. 2,3 and which has no (C 5 K 2 )-component, then 8i(G) ≤ 8n 0 (G) + 5n 1 (G) + 4n 2 (G) + 3n 3 (G).…”

“…If G = C 5 K 2 is a connected cubic graph of order n that does not have a subgraph isomorphic to K 2,3 , then i(G) ≤ 3n/8.…”

“…Note also that the condition in Theorem 3 that G does not contain an induced K 2,3 cannot be dropped. Indeed, the graph obtained by attaching a leaf to a degree 2 vertex of K 2,3 (or equivalently by deleting two adjacent edges in K 3,3 ) has no independent dominating set of size less than three. Therefore, 8i(G)…”

Distance-2 Independent Domination Numbers

“…The theory of independent domination was proposed by Berge [2] and Ore [8] in 1962. The independent domination number and the notation i(G) were introduced by Cockayne and Hedetniemi in [3,4]. Recently, it was then extensively studied for various classes of graphs in the literature (see [5], [6], [7], [9], [10], [11]).…”