288 research outputs found

    A simple linear time algorithm for the locally connected spanning tree problem on maximal planar chordal graphs

    Get PDF
    A locally connected spanning tree (LCST) T of a graph G is a spanning tree of G such that, for each node, its neighborhood in T induces a connected sub- graph in G. The problem of determining whether a graph contains an LCST or not has been proved to be NP-complete, even if the graph is planar or chordal. The main result of this paper is a simple linear time algorithm that, given a maximal planar chordal graph, determines in linear time whether it contains an LCST or not, and produces one if it exists. We give an anal- ogous result for the case when the input graph is a maximal outerplanar graph

    On Computing the Average Distance for Some Chordal-Like Graphs

    Get PDF
    The Wiener index of a graph G is the sum of all its distances. Up to renormalization, it is also the average distance in G. The problem of computing this parameter has different applications in chemistry and networks. We here study when it can be done in truly subquadratic time (in the size n+m of the input) on n-vertex m-edge graphs. Our main result is a complete answer to this question, assuming the Strong Exponential-Time Hypothesis (SETH), for all the hereditary subclasses of chordal graphs. Interestingly, the exact same result also holds for the diameter problem. The case of non-hereditary chordal subclasses happens to be more challenging. For the chordal Helly graphs we propose an intricate O?(m^{3/2})-time algorithm for computing the Wiener index, where m denotes the number of edges. We complete our results with the first known linear-time algorithm for this problem on the dually chordal graphs. The former algorithm also computes the median set

    Defective Coloring on Classes of Perfect Graphs

    Full text link
    In Defective Coloring we are given a graph GG and two integers χd\chi_d, Δ\Delta^* and are asked if we can χd\chi_d-color GG so that the maximum degree induced by any color class is at most Δ\Delta^*. We show that this natural generalization of Coloring is much harder on several basic graph classes. In particular, we show that it is NP-hard on split graphs, even when one of the two parameters χd\chi_d, Δ\Delta^* is set to the smallest possible fixed value that does not trivialize the problem (χd=2\chi_d = 2 or Δ=1\Delta^* = 1). Together with a simple treewidth-based DP algorithm this completely determines the complexity of the problem also on chordal graphs. We then consider the case of cographs and show that, somewhat surprisingly, Defective Coloring turns out to be one of the few natural problems which are NP-hard on this class. We complement this negative result by showing that Defective Coloring is in P for cographs if either χd\chi_d or Δ\Delta^* is fixed; that it is in P for trivially perfect graphs; and that it admits a sub-exponential time algorithm for cographs when both χd\chi_d and Δ\Delta^* are unbounded

    On Minimum Maximal Distance-k Matchings

    Full text link
    We study the computational complexity of several problems connected with finding a maximal distance-kk matching of minimum cardinality or minimum weight in a given graph. We introduce the class of kk-equimatchable graphs which is an edge analogue of kk-equipackable graphs. We prove that the recognition of kk-equimatchable graphs is co-NP-complete for any fixed k2k \ge 2. We provide a simple characterization for the class of strongly chordal graphs with equal kk-packing and kk-domination numbers. We also prove that for any fixed integer 1\ell \ge 1 the problem of finding a minimum weight maximal distance-22\ell matching and the problem of finding a minimum weight (21)(2 \ell - 1)-independent dominating set cannot be approximated in polynomial time in chordal graphs within a factor of δlnV(G)\delta \ln |V(G)| unless P=NP\mathrm{P} = \mathrm{NP}, where δ\delta is a fixed constant (thereby improving the NP-hardness result of Chang for the independent domination case). Finally, we show the NP-hardness of the minimum maximal induced matching and independent dominating set problems in large-girth planar graphs.Comment: 15 pages, 4 figure

    Equivalence of the filament and overlap graphs of subtrees of limited trees

    Get PDF
    The overlap graphs of subtrees of a tree are equivalent to subtree filament graphs, the overlap graphs of subtrees of a star are cocomparability graphs, and the overlap graphs of subtrees of a caterpillar are interval filament graphs. In this paper, we show the equivalence of many more classes of subtree overlap and subtree filament graphs, and equate them to classes of complements of cochordal-mixed graphs. Our results generalize the previously known results mentioned above
    corecore