216 research outputs found
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
On Hypergraph Supports
Let be a hypergraph. A support is a graph
on such that for each , the subgraph of induced on the
elements in is connected. In this paper, we consider hypergraphs defined on
a host graph. Given a graph , with ,
and a collection of connected subgraphs of , a primal support
is a graph on such that for each , the
induced subgraph on vertices is connected. A \emph{dual support} is a graph on
s.t. for each , the induced subgraph
is connected, where . 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: If the host graph has genus
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
. If the host graph has treewidth and the hyperedges satisfy a
combinatorial condition of being \emph{non-piercing}, then there exist primal
and dual supports of treewidth . 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
LIPIcs, Volume 274, ESA 2023, Complete Volum
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
Succinct and Compact Data Structures for Intersection Graphs
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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