2,656 research outputs found
Induced Subgraph Isomorphism on proper interval and bipartite permutation graphs *
Abstract Given two graphs G and H as input, the Induced Subgraph Isomorphism (ISI) problem is to decide whether G has an induced subgraph that is isomorphic to H. This problem is NP-complete already when G and H are restricted to disjoint unions of paths, and consequently also NP-complete on proper interval graphs and on bipartite permutation graphs. We show that ISI can be solved in polynomial time on proper interval graphs and on bipartite permutation graphs, provided that H is connected. As a consequence, we obtain that ISI is fixed-parameter tractable on these two graph classes, when parametrised by the number of connected components of H. Our results contrast and complement the following known results: W [1]-hardness of ISI on interval graphs when parametrised by the number of vertices of H, NP-completeness of ISI on connected interval graphs and on connected permutation graphs, and NP-completeness of Subgraph Isomorphism on connected proper interval graphs and connected bipartite permutation graphs
Graph Isomorphism for unit square graphs
In the past decades for more and more graph classes the Graph Isomorphism
Problem was shown to be solvable in polynomial time. An interesting family of
graph classes arises from intersection graphs of geometric objects. In this
work we show that the Graph Isomorphism Problem for unit square graphs,
intersection graphs of axis-parallel unit squares in the plane, can be solved
in polynomial time. Since the recognition problem for this class of graphs is
NP-hard we can not rely on standard techniques for geometric graphs based on
constructing a canonical realization. Instead, we develop new techniques which
combine structural insights into the class of unit square graphs with
understanding of the automorphism group of such graphs. For the latter we
introduce a generalization of bounded degree graphs which is used to capture
the main structure of unit square graphs. Using group theoretic algorithms we
obtain sufficient information to solve the isomorphism problem for unit square
graphs.Comment: 31 pages, 6 figure
Modular Decomposition and the Reconstruction Conjecture
We prove that a large family of graphs which are decomposable with respect to
the modular decomposition can be reconstructed from their collection of
vertex-deleted subgraphs.Comment: 9 pages, 2 figure
Combinatorial Problems on -graphs
Bir\'{o}, Hujter, and Tuza introduced the concept of -graphs (1992),
intersection graphs of connected subgraphs of a subdivision of a graph .
They naturally generalize many important classes of graphs, e.g., interval
graphs and circular-arc graphs. We continue the study of these graph classes by
considering coloring, clique, and isomorphism problems on -graphs.
We show that for any fixed containing a certain 3-node, 6-edge multigraph
as a minor that the clique problem is APX-hard on -graphs and the
isomorphism problem is isomorphism-complete. We also provide positive results
on -graphs. Namely, when is a cactus the clique problem can be solved in
polynomial time. Also, when a graph has a Helly -representation, the
clique problem can be solved in polynomial time. Finally, we observe that one
can use treewidth techniques to show that both the -clique and list
-coloring problems are FPT on -graphs. These FPT results apply more
generally to treewidth-bounded graph classes where treewidth is bounded by a
function of the clique number
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