26 research outputs found
Solving the Canonical Representation and Star System Problems for Proper Circular-Arc Graphs in Log-Space
We present a logspace algorithm that constructs a canonical intersection
model for a given proper circular-arc graph, where `canonical' means that
models of isomorphic graphs are equal. This implies that the recognition and
the isomorphism problems for this class of graphs are solvable in logspace. For
a broader class of concave-round graphs, that still possess (not necessarily
proper) circular-arc models, we show that those can also be constructed
canonically in logspace. As a building block for these results, we show how to
compute canonical models of circular-arc hypergraphs in logspace, which are
also known as matrices with the circular-ones property. Finally, we consider
the search version of the Star System Problem that consists in reconstructing a
graph from its closed neighborhood hypergraph. We solve it in logspace for the
classes of proper circular-arc, concave-round, and co-convex graphs.Comment: 19 pages, 3 figures, major revisio
Deciding Circular-Arc Graph Isomorphism in Parameterized Logspace
We compute a canonical circular-arc representation for a given circular-arc
(CA) graph which implies solving the isomorphism and recognition problem for
this class. To accomplish this we split the class of CA graphs into uniform and
non-uniform ones and employ a generalized version of the argument given by
K\"obler et al (2013) that has been used to show that the subclass of Helly CA
graphs can be canonized in logspace. For uniform CA graphs our approach works
in logspace and in addition to that Helly CA graphs are a strict subset of
uniform CA graphs. Thus our result is a generalization of the canonization
result for Helly CA graphs. In the non-uniform case a specific set of ambiguous
vertices arises. By choosing the parameter to be the cardinality of this set
the obstacle can be solved by brute force. This leads to an O(k + log n) space
algorithm to compute a canonical representation for non-uniform and therefore
all CA graphs.Comment: 14 pages, 3 figure
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