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- Title
ON THE CO-ROMAN DOMINATION IN GRAPHS.
- Authors
ZEHUI SHAO; SHEIKHOLESLAMI, SEYED MAHMOUD; SOROUDI, MARZIEH; VOLKMANN, LUTZ; XINMIAO LIU
- Abstract
Let G = (V,E) be a graph and let f : V (G) → {0, 1, 2} be a function. A vertex v is said to be protected with respect to f, if f(v) > 0 or f(v) = 0 and v is adjacent to a vertex of positive weight. The function f is a co-Roman dominating function if (i) every vertex in V is protected, and (ii) each v ∊ V with positive weight has a neighbor u ∊ V with f(u) = 0 such that the function fuv : V → {0, 1, 2}, defined by fuv(u) = 1, fuv(v) = f(v) - 1 and fuv(x) = f(x) for x ∊ V \ {v, u}, has no unprotected vertex. The weight of f is w(f) = ∑v∊V f(v). The co-Roman domination number of a graph G, denoted by cr(G), is the minimum weight of a co-Roman dominating function on G. In this paper, we give a characterization of graphs of order n for which co-Roman domination number is 2n/3 or n - 2, which settlestwo open problem in [S. Arumugam, K. Ebadi and M. Manrique, Co-Roman domination in graphs, Proc. Indian Acad. Sci. Math. Sci. 125 (2015) 1-10]. Furthermore, we present some sharp bounds on the co-Roman domination number.
- Subjects
GRAPH connectivity; ISOMORPHISM (Mathematics); SUBGRAPHS; BIPARTITE graphs; ROMAN numerals
- Publication
Discussiones Mathematicae: Graph Theory, 2019, Vol 39, Issue 2, p455
- ISSN
1234-3099
- Publication type
Article
- DOI
10.7151/dmgt.2091