216 research outputs found

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    On Hypergraph Supports

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    Let H=(X,E)\mathcal{H}=(X,\mathcal{E}) be a hypergraph. A support is a graph QQ on XX such that for each E∈EE\in\mathcal{E}, the subgraph of QQ induced on the elements in EE is connected. In this paper, we consider hypergraphs defined on a host graph. Given a graph G=(V,E)G=(V,E), with c:V→{r,b}c:V\to\{\mathbf{r},\mathbf{b}\}, and a collection of connected subgraphs H\mathcal{H} of GG, a primal support is a graph QQ on b(V)\mathbf{b}(V) such that for each H∈HH\in \mathcal{H}, the induced subgraph Q[b(H)]Q[\mathbf{b}(H)] on vertices b(H)=H∩c−1(b)\mathbf{b}(H)=H\cap c^{-1}(\mathbf{b}) is connected. A \emph{dual support} is a graph Q∗Q^* on H\mathcal{H} s.t. for each v∈Xv\in X, the induced subgraph Q∗[Hv]Q^*[\mathcal{H}_v] is connected, where Hv={H∈H:v∈H}\mathcal{H}_v=\{H\in\mathcal{H}: v\in H\}. We present sufficient conditions on the host graph and hyperedges so that the resulting support comes from a restricted family. We primarily study two classes of graphs: (1)(1) If the host graph has genus gg and the hypergraphs satisfy a topological condition of being \emph{cross-free}, then there is a primal and a dual support of genus at most gg. (2)(2) If the host graph has treewidth tt and the hyperedges satisfy a combinatorial condition of being \emph{non-piercing}, then there exist primal and dual supports of treewidth O(2t)O(2^t). We show that this exponential blow-up is sometimes necessary. As an intermediate case, we also study the case when the host graph is outerplanar. Finally, we show applications of our results to packing and covering, and coloring problems on geometric hypergraphs

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum

    Succinct and Compact Data Structures for Intersection Graphs

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    This thesis designs space efficient data structures for several classes of intersection graphs, including interval graphs, path graphs and chordal graphs. Our goal is to support navigational operations such as adjacent and neighbourhood and distance operations such as distance efficiently while occupying optimal space, or a constant factor of the optimal space. Using our techniques, we first resolve an open problem with regards to succinctly representing ordinal trees that is able to convert between the index of a node in a depth-first traversal (i.e. pre-order) and in a breadth-first traversal (i.e. level-order) of the tree. Using this, we are able to augment previous succinct data structures for interval graphs with the \GDistance operation. We also study several variations of the data structure problem in interval graphs: beer interval graphs and dynamic interval graphs. In beer interval graphs, we are given that some vertices of the graph are beer nodes (representing beer stores) and we consider only those paths that pass through at least one of these beer nodes. We give data structure results and prove space lower bounds for these graphs. We study dynamic interval graphs under several well known dynamic models such as incremental or offline, and we give data structures for each of these models. Finally we consider path graphs where we improve on previous works by exploiting orthogonal range reporting data structures. For optimal space representations, we improve the run time of the queries, while for non-optimal space representations (but optimal query times), we reduce the space needed
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