26,982 research outputs found

    Line-distortion, Bandwidth and Path-length of a graph

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    We investigate the minimum line-distortion and the minimum bandwidth problems on unweighted graphs and their relations with the minimum length of a Robertson-Seymour's path-decomposition. The length of a path-decomposition of a graph is the largest diameter of a bag in the decomposition. The path-length of a graph is the minimum length over all its path-decompositions. In particular, we show: - if a graph GG can be embedded into the line with distortion kk, then GG admits a Robertson-Seymour's path-decomposition with bags of diameter at most kk in GG; - for every class of graphs with path-length bounded by a constant, there exist an efficient constant-factor approximation algorithm for the minimum line-distortion problem and an efficient constant-factor approximation algorithm for the minimum bandwidth problem; - there is an efficient 2-approximation algorithm for computing the path-length of an arbitrary graph; - AT-free graphs and some intersection families of graphs have path-length at most 2; - for AT-free graphs, there exist a linear time 8-approximation algorithm for the minimum line-distortion problem and a linear time 4-approximation algorithm for the minimum bandwidth problem

    Structural parameterizations for boxicity

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    The boxicity of a graph GG is the least integer dd such that GG has an intersection model of axis-aligned dd-dimensional boxes. Boxicity, the problem of deciding whether a given graph GG has boxicity at most dd, is NP-complete for every fixed d≥2d \ge 2. We show that boxicity is fixed-parameter tractable when parameterized by the cluster vertex deletion number of the input graph. This generalizes the result of Adiga et al., that boxicity is fixed-parameter tractable in the vertex cover number. Moreover, we show that boxicity admits an additive 11-approximation when parameterized by the pathwidth of the input graph. Finally, we provide evidence in favor of a conjecture of Adiga et al. that boxicity remains NP-complete when parameterized by the treewidth.Comment: 19 page

    Bandwidth theorem for random graphs

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    A graph GG is said to have \textit{bandwidth} at most bb, if there exists a labeling of the vertices by 1,2,...,n1,2,..., n, so that ∣i−j∣≤b|i - j| \leq b whenever {i,j}\{i,j\} is an edge of GG. Recently, B\"{o}ttcher, Schacht, and Taraz verified a conjecture of Bollob\'{a}s and Koml\'{o}s which says that for every positive r,Δ,γr,\Delta,\gamma, there exists β\beta such that if HH is an nn-vertex rr-chromatic graph with maximum degree at most Δ\Delta which has bandwidth at most βn\beta n, then any graph GG on nn vertices with minimum degree at least (1−1/r+γ)n(1 - 1/r + \gamma)n contains a copy of HH for large enough nn. In this paper, we extend this theorem to dense random graphs. For bipartite HH, this answers an open question of B\"{o}ttcher, Kohayakawa, and Taraz. It appears that for non-bipartite HH the direct extension is not possible, and one needs in addition that some vertices of HH have independent neighborhoods. We also obtain an asymptotically tight bound for the maximum number of vertex disjoint copies of a fixed rr-chromatic graph H0H_0 which one can find in a spanning subgraph of G(n,p)G(n,p) with minimum degree (1−1/r+γ)np(1-1/r + \gamma)np.Comment: 29 pages, 3 figure

    Bandwidth and density for block graphs

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    The bandwidth of a graph G is the minimum of the maximum difference between adjacent labels when the vertices have distinct integer labels. We provide a polynomial algorithm to produce an optimal bandwidth labeling for graphs in a special class of block graphs (graphs in which every block is a clique), namely those where deleting the vertices of degree one produces a path of cliques. The result is best possible in various ways. Furthermore, for two classes of graphs that are ``almost'' caterpillars, the bandwidth problem is NP-complete.Comment: 14 pages, 9 included figures. Note: figures did not appear in original upload; resubmission corrects thi
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