643 research outputs found
Upper bounds for alpha-domination parameters
In this paper, we provide a new upper bound for the alpha-domination number.
This result generalises the well-known Caro-Roditty bound for the domination
number of a graph. The same probabilistic construction is used to generalise
another well-known upper bound for the classical domination in graphs. We also
prove similar upper bounds for the alpha-rate domination number, which combines
the concepts of alpha-domination and k-tuple domination.Comment: 7 pages; Presented at the 4th East Coast Combinatorial Conference,
Antigonish (Nova Scotia, Canada), May 1-2, 200
3-Factor-criticality in double domination edge critical graphs
A vertex subset of a graph is a double dominating set of if
for each vertex of , where is the set of the
vertex and vertices adjacent to . The double domination number of ,
denoted by , is the cardinality of a smallest double
dominating set of . A graph is said to be double domination edge
critical if for any edge . A double domination edge critical graph with is called --critical. A graph is
-factor-critical if has a perfect matching for each set of
vertices in . In this paper we show that is 3-factor-critical if is
a 3-connected claw-free --critical graph of odd order
with minimum degree at least 4 except a family of graphs.Comment: 14 page
Locating and Total Dominating Sets of Direct Products of Complete Graphs
A set S of vertices in a graph G = (V,E) is a metric-locating-total dominating set of G if every vertex of V is adjacent to a vertex in S and for every u ≠v in V there is a vertex x in S such that d(u,x) ≠d(v,x). The metric-location-total domination number \gamma^M_t(G) of G is the minimum cardinality of a metric-locating-total dominating set in G. For graphs G and H, the direct product G × H is the graph with vertex set V(G) × V(H) where two vertices (x,y) and (v,w) are adjacent if and only if xv in E(G) and yw in E(H). In this paper, we determine the lower bound of the metric-location-total domination number of the direct products of complete graphs. We also determine some exact values for some direct products of two complete graphs
The k-tuple domination number revisited
The following fundamental result for the domination number γ (G) of a graph G was proved by Alon and Spencer, Arnautov, Lovász and Payan: γ (G) ≤ frac(ln (δ + 1) + 1, δ + 1) n, where n is the order and δ is the minimum degree of vertices of G. A similar upper bound for the double domination number was found by Harant and Henning [J. Harant, M.A. Henning, On double domination in graphs, Discuss. Math. Graph Theory 25 (2005) 29-34], and for the triple domination number by Rautenbach and Volkmann [D. Rautenbach, L. Volkmann, New bounds on the k-domination number and the k-tuple domination number, Appl. Math. Lett. 20 (2007) 98-102], who also posed the interesting conjecture on the k-tuple domination number: for any graph G with δ ≥ k - 1, γ× k (G) ≤ frac(ln (δ - k + 2) + ln (over(d, ̂)k - 1 + over(d, ̂)k - 2) + 1, δ - k + 2) n, where over(d, ̂)m = ∑i = 1n ((di; m)) / n is the m-degree of G. This conjecture, if true, would generalize all the mentioned upper bounds and improve an upper bound proved in [A. Gagarin, V. Zverovich, A generalised upper bound for the k-tuple domination number, Discrete Math. (2007), in press (doi:10.1016/j.disc.2007.07.033)]. In this paper, we prove the Rautenbach-Volkmann conjecture. © 2007 Elsevier Ltd. All rights reserved
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