1,259 research outputs found

    Disjunctive Total Domination in Graphs

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    Let GG be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, γt(G)\gamma_t(G). A set SS of vertices in GG is a disjunctive total dominating set of GG if every vertex is adjacent to a vertex of SS or has at least two vertices in SS at distance2 from it. The disjunctive total domination number, γtd(G)\gamma^d_t(G), is the minimum cardinality of such a set. We observe that γtd(G)γt(G)\gamma^d_t(G) \le \gamma_t(G). We prove that if GG is a connected graph of ordern8n \ge 8, then γtd(G)2(n1)/3\gamma^d_t(G) \le 2(n-1)/3 and we characterize the extremal graphs. It is known that if GG is a connected claw-free graph of ordernn, then γt(G)2n/3\gamma_t(G) \le 2n/3 and this upper bound is tight for arbitrarily largenn. We show this upper bound can be improved significantly for the disjunctive total domination number. We show that if GG is a connected claw-free graph of ordern>10n > 10, then γtd(G)4n/7\gamma^d_t(G) \le 4n/7 and we characterize the graphs achieving equality in this bound.Comment: 23 page

    Algorithmic aspects of disjunctive domination in graphs

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    For a graph G=(V,E)G=(V,E), a set DVD\subseteq V is called a \emph{disjunctive dominating set} of GG if for every vertex vVDv\in V\setminus D, vv is either adjacent to a vertex of DD or has at least two vertices in DD at distance 22 from it. The cardinality of a minimum disjunctive dominating set of GG is called the \emph{disjunctive domination number} of graph GG, and is denoted by γ2d(G)\gamma_{2}^{d}(G). The \textsc{Minimum Disjunctive Domination Problem} (MDDP) is to find a disjunctive dominating set of cardinality γ2d(G)\gamma_{2}^{d}(G). Given a positive integer kk and a graph GG, the \textsc{Disjunctive Domination Decision Problem} (DDDP) is to decide whether GG has a disjunctive dominating set of cardinality at most kk. In this article, we first propose a linear time algorithm for MDDP in proper interval graphs. Next we tighten the NP-completeness of DDDP by showing that it remains NP-complete even in chordal graphs. We also propose a (ln(Δ2+Δ+2)+1)(\ln(\Delta^{2}+\Delta+2)+1)-approximation algorithm for MDDP in general graphs and prove that MDDP can not be approximated within (1ϵ)ln(V)(1-\epsilon) \ln(|V|) for any ϵ>0\epsilon>0 unless NP \subseteq DTIME(VO(loglogV))(|V|^{O(\log \log |V|)}). Finally, we show that MDDP is APX-complete for bipartite graphs with maximum degree 33

    Space-Time Tradeoffs for Conjunctive Queries with Access Patterns

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    In this paper, we investigate space-time tradeoffs for answering conjunctive queries with access patterns (CQAPs). The goal is to create a space-efficient data structure in an initial preprocessing phase and use it for answering (multiple) queries in an online phase. Previous work has developed data structures that trades off space usage for answering time for queries of practical interest, such as the path and triangle query. However, these approaches lack a comprehensive framework and are not generalizable. Our main contribution is a general algorithmic framework for obtaining space-time tradeoffs for any CQAP. Our framework builds upon the \PANDA algorithm and tree decomposition techniques. We demonstrate that our framework captures all state-of-the-art tradeoffs that were independently produced for various queries. Further, we show surprising improvements over the state-of-the-art tradeoffs known in the existing literature for reachability queries

    Independent transversal total domination versus total domination in trees

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    A subset of vertices in a graph G is a total dominating set if every vertex in G is adjacent to at least one vertex in this subset. The total domination number of G is the minimum cardinality of any total dominating set in G and is denoted by gamma(t)(G). A total dominating set of G having nonempty intersection with all the independent sets of maximum cardinality in G is an independent transversal total dominating set. The minimum cardinality of any independent transversal total dominating set is denoted by gamma(u) (G). Based on the fact that for any tree T, gamma(t) (T) <= gamma(u) (T) <= gamma(t) (T) + 1, in this work we give several relationship(s) between gamma(u) (T) and gamma(t) (T) for trees T which are leading to classify the trees which are satisfying the equality in these bound
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