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Let G = (V,E) be a graph and k be a positive integer. A k- dominating set of G is a subset S ⊆ V such that each vertex in V＼S has at least k neighbors in S. A Roman k-dominating function on G is a function f : V ! {0, 1, 2} such that every vertex v with f(v) = 0 is adjacent to at least k vertices v1, v2, . . . , vk with f(vi) = 2 for i = 1, 2, . . . , k. In the paper titled “Roman k-domination in graphs” (J. Korean Math. Soc. 46 (2009), no. 6, 1309–1318) K. Kammerling and L. Volkmann showed that for any graph G with n vertices, kR(G) + kR(G) min {2n, 4k + 1}, and the equality holds if and only if n 2k or k 2 and n = 2k + 1 or k = 1 and G or G has a vertex of degree n − 1 and its complement has a vertex of degree n − 2. In this paper we find a counterexample of Kammerling and Volkmann’s result and then give a correction to the result.
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참고문헌 (5)
참고문헌 신청
C. S. Liao / 2002 / Algorithmic aspect of k-tuple domination in graphs / Taiwanese J. Math 6 (3) : 415 ~ 420
C. S. ReVelle / 2000 / Defendens imperium Romanum: a classical problem in military strategy / Amer. Math. Monthly 107 (7) : 585 ~ 594
Karsten Kammerling / 2009 / Roman k-domination in graphs / 대한수학회지 46 (6) : 1309 ~ 1318
T. W. Haynes / 1998 / Fundamentals of Domination in Graphs / Marcel Dekker
W. Shang / 2008 / On minimum m-connected k-dominating set problem in unit disc graphs / J. Comb. Optim 16 (2) : 99 ~ 106
C. S. ReVelle / 2000 / Defendens imperium Romanum: a classical problem in military strategy / Amer. Math. Monthly 107 (7) : 585 ~ 594
Karsten Kammerling / 2009 / Roman k-domination in graphs / 대한수학회지 46 (6) : 1309 ~ 1318
T. W. Haynes / 1998 / Fundamentals of Domination in Graphs / Marcel Dekker
W. Shang / 2008 / On minimum m-connected k-dominating set problem in unit disc graphs / J. Comb. Optim 16 (2) : 99 ~ 106
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