2 research outputs found

    Edge-Vertex Dominating Set in Unit Disk Graphs

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    Given an undirected graph G=(V,E)G=(V,E), a vertex v∈Vv\in V is edge-vertex (ev) dominated by an edge e∈Ee\in E if vv is either incident to ee or incident to an adjacent edge of ee. A set SevβŠ†ES^{ev}\subseteq E is an edge-vertex dominating set (referred to as ev-dominating set) of GG if every vertex of GG is ev-dominated by at least one edge of SevS^{ev}. The minimum cardinality of an ev-dominating set is the ev-domination number. The edge-vertex dominating set problem is to find a minimum ev-domination number. In this paper we prove that the ev-dominating set problem is {\tt NP-hard} on unit disk graphs. We also prove that this problem admits a polynomial-time approximation scheme on unit disk graphs. Finally, we give a simple 5-factor linear-time approximation algorithm

    Vertex-edge perfect Roman domination number

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    A vertex-edge perfect Roman dominating function on a graph G=(V,E) G = (V, E) (denoted by ve-PRDF) is a function f:V(G)⟢{0,1,2} f:V\left(G\right)\longrightarrow\{0, 1, 2\} such that for every edge uv∈E uv\in E , max⁑{f(u),f(v)}β‰ 0 \max\{f(u), f(v)\}\neq0 , or u u is adjacent to exactly one neighbor w w such that f(w)=2 f(w) = 2 , or v v is adjacent to exactly one neighbor w w such that f(w)=2 f(w) = 2 . The weight of a ve-PRDF on G G is the sum w(f)=βˆ‘v∈Vf(v) w(f) = \sum_{v\in V}f(v) . The vertex-edge perfect Roman domination number of G G (denoted by Ξ³veRp(G) \gamma_{veR}^{p}(G) ) is the minimum weight of a ve-PRDF on G G . In this paper, we first show that vertex-edge perfect Roman dominating is NP-complete for bipartite graphs. Also, for a tree T T , we give upper and lower bounds for Ξ³veRp(T) \gamma_{veR}^{p}(T) in terms of the order n n , l l leaves and s s support vertices. Lastly, we determine Ξ³veRp(G) \gamma_{veR}^{p}(G) for Petersen, cycle and Flower snark graphs
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