19,983 research outputs found

    Domination Cover Pebbling: Structural Results

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    This paper continues the results of "Domination Cover Pebbling: Graph Families." An almost sharp bound for the domination cover pebbling (DCP) number for graphs G with specified diameter has been computed. For graphs of diameter two, a bound for the ratio between the cover pebbling number of G and the DCP number of G has been computed. A variant of domination cover pebbling, called subversion DCP is introducted, and preliminary results are discussed.Comment: 15 page

    Computing Minimum Rainbow and Strong Rainbow Colorings of Block Graphs

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    A path in an edge-colored graph GG is rainbow if no two edges of it are colored the same. The graph GG is rainbow-connected if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair of vertices, the graph GG is strongly rainbow-connected. The minimum number of colors needed to make GG rainbow-connected is known as the rainbow connection number of GG, and is denoted by rc(G)\text{rc}(G). Similarly, the minimum number of colors needed to make GG strongly rainbow-connected is known as the strong rainbow connection number of GG, and is denoted by src(G)\text{src}(G). We prove that for every k≥3k \geq 3, deciding whether src(G)≤k\text{src}(G) \leq k is NP-complete for split graphs, which form a subclass of chordal graphs. Furthermore, there exists no polynomial-time algorithm for approximating the strong rainbow connection number of an nn-vertex split graph with a factor of n1/2−ϵn^{1/2-\epsilon} for any ϵ>0\epsilon > 0 unless P = NP. We then turn our attention to block graphs, which also form a subclass of chordal graphs. We determine the strong rainbow connection number of block graphs, and show it can be computed in linear time. Finally, we provide a polynomial-time characterization of bridgeless block graphs with rainbow connection number at most 4.Comment: 13 pages, 3 figure

    All Maximal Independent Sets and Dynamic Dominance for Sparse Graphs

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    We describe algorithms, based on Avis and Fukuda's reverse search paradigm, for listing all maximal independent sets in a sparse graph in polynomial time and delay per output. For bounded degree graphs, our algorithms take constant time per set generated; for minor-closed graph families, the time is O(n) per set, and for more general sparse graph families we achieve subquadratic time per set. We also describe new data structures for maintaining a dynamic vertex set S in a sparse or minor-closed graph family, and querying the number of vertices not dominated by S; for minor-closed graph families the time per update is constant, while it is sublinear for any sparse graph family. We can also maintain a dynamic vertex set in an arbitrary m-edge graph and test the independence of the maintained set in time O(sqrt m) per update. We use the domination data structures as part of our enumeration algorithms.Comment: 10 page
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