1,762 research outputs found

    k-Tuple_Total_Domination_in_Inflated_Graphs

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    The inflated graph GIG_{I} of a graph GG with n(G)n(G) vertices is obtained from GG by replacing every vertex of degree dd of GG by a clique, which is isomorph to the complete graph KdK_{d}, and each edge (xi,xj)(x_{i},x_{j}) of GG is replaced by an edge (u,v)(u,v) in such a way that u∈Xiu\in X_{i}, v∈Xjv\in X_{j}, and two different edges of GG are replaced by non-adjacent edges of GIG_{I}. For integer kβ‰₯1k\geq 1, the kk-tuple total domination number Ξ³Γ—k,t(G)\gamma_{\times k,t}(G) of GG is the minimum cardinality of a kk-tuple total dominating set of GG, which is a set of vertices in GG such that every vertex of GG is adjacent to at least kk vertices in it. For existing this number, must the minimum degree of GG is at least kk. Here, we study the kk-tuple total domination number in inflated graphs when kβ‰₯2k\geq 2. First we prove that n(G)k≀γ×k,t(GI)≀n(G)(k+1)βˆ’1n(G)k\leq \gamma_{\times k,t}(G_{I})\leq n(G)(k+1)-1, and then we characterize graphs GG that the kk-tuple total domination number number of GIG_I is n(G)kn(G)k or n(G)k+1n(G)k+1. Then we find bounds for this number in the inflated graph GIG_I, when GG has a cut-edge ee or cut-vertex vv, in terms on the kk-tuple total domination number of the inflated graphs of the components of Gβˆ’eG-e or vv-components of Gβˆ’vG-v, respectively. Finally, we calculate this number in the inflated graphs that have obtained by some of the known graphs

    {Linear Kernels for kk-Tupel and Liar's Domination in Bounded Genus Graphs}

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    A set DβŠ†VD\subseteq V is called a kk-tuple dominating set of a graph G=(V,E)G=(V,E) if ∣NG[v]∩D∣β‰₯k\left| N_G[v] \cap D \right| \geq k for all v∈Vv \in V, where NG[v]N_G[v] denotes the closed neighborhood of vv. A set DβŠ†VD \subseteq V is called a liar's dominating set of a graph G=(V,E)G=(V,E) if (i) ∣NG[v]∩D∣β‰₯2\left| N_G[v] \cap D \right| \geq 2 for all v∈Vv\in V and (ii) for every pair of distinct vertices u,v∈Vu, v\in V, ∣(NG[u]βˆͺNG[v])∩D∣β‰₯3\left| (N_G[u] \cup N_G[v]) \cap D \right| \geq 3. Given a graph GG, the decision versions of kk-Tuple Domination Problem and the Liar's Domination Problem are to check whether there exists a kk-tuple dominating set and a liar's dominating set of GG of a given cardinality, respectively. These two problems are known to be NP-complete \cite{LiaoChang2003, Slater2009}. In this paper, we study the parameterized complexity of these problems. We show that the kk-Tuple Domination Problem and the Liar's Domination Problem are W[2]\mathsf{W}[2]-hard for general graphs but they admit linear kernels for graphs with bounded genus.Title changed from "Parameterized complexity of k-tuple and liar's domination" to "Linear kernels for k-tuple and liar's domination in bounded genus graphs

    Upper bounds for alpha-domination parameters

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    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
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