Abstract A set S of vertices in a graph G is a dominating set if every vertex not in S is adjacent to a vertex in S . If, in addition, S is an independent set, then S is an independent dominating set. The domination number γ ( G ) of G is the minimum cardinality of a dominating set in G , while the independent domination number i ( G ) of G is the minimum cardinality of an independent dominating set in G . It is known (Goddard et al., 2012) that if G is a connected 3-regular graph, then i ( G ) ∕ γ ( G ) ≤ 3 ∕ 2 , with equality if and only if G = K 3 , 3 . In this paper, we extend this result to graphs of larger regularity and show that if k ∈ { 4 , 5 , 6 } and G is a connected k -regular graph, then i ( G ) ∕ γ ( G ) ≤ k ∕ 2 , with equality if and only if G = K k , k .

中文翻译：

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小正则图中的支配与独立支配

摘要 如果每个不在 S 中的顶点都与 S 中的一个顶点相邻，则图 G 中的顶点集 S 是支配集。此外，如果 S 是独立集，则 S 是独立支配集。G的支配数γ(G)是G中支配集的最小基数，而G的独立支配数i(G)是G中独立支配集的最小基数。已知 (Goddard et al., 2012) 如果 G 是连通的 3-正则图，则 i ( G ) ∕ γ ( G ) ≤ 3 ∕ 2 ，当且仅当 G = K 3 , 3 时相等。在本文中，我们将这个结果扩展到具有更大正则性的图，并证明如果 k ∈ { 4 , 5 , 6 } 并且 G 是连通的 k -正则图，则 i ( G ) ∕ γ ( G ) ≤ k ∕ 2 , 相等当且仅当 G = K k , k 。