Isomorphism of graph classes related to the circular-ones property

We give a linear-time algorithm that checks for isomorphism between two 0-1 matrices that obey the circular-ones property. This algorithm leads to linear-time isomorphism algorithms for related graph classes, including Helly circular-arc graphs, \Gamma-circular-arc graphs, proper circular-arc graphs and convex-round graphs.Comment: 25 pages, 9 figure

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

Combinatorial Problems on HH-graphs

Bir\'{o}, Hujter, and Tuza introduced the concept of $H$-graphs (1992), intersection graphs of connected subgraphs of a subdivision of a graph $H$. 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 $H$-graphs. We show that for any fixed $H$ containing a certain 3-node, 6-edge multigraph as a minor that the clique problem is APX-hard on $H$-graphs and the isomorphism problem is isomorphism-complete. We also provide positive results on $H$-graphs. Namely, when $H$ is a cactus the clique problem can be solved in polynomial time. Also, when a graph $G$ has a Helly $H$-representation, the clique problem can be solved in polynomial time. Finally, we observe that one can use treewidth techniques to show that both the $k$-clique and list $k$-coloring problems are FPT on $H$-graphs. These FPT results apply more generally to treewidth-bounded graph classes where treewidth is bounded by a function of the clique number

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 completeness for trapezoid graphs

The complexity of the graph isomorphism problem for trapezoid graphs has been open over a decade. This paper shows that the problem is GI-complete. More precisely, we show that the graph isomorphism problem is GI-complete for comparability graphs of partially ordered sets with interval dimension 2 and height 3. In contrast, the problem is known to be solvable in polynomial time for comparability graphs of partially ordered sets with interval dimension at most 2 and height at most 2.Comment: 4 pages, 3 Postscript figure

Dimers and cluster integrable systems

We show that the dimer model on a bipartite graph on a torus gives rise to a quantum integrable system of special type - a cluster integrable system. The phase space of the classical system contains, as an open dense subset, the moduli space of line bundles with connections on the graph. The sum of Hamiltonians is essentially the partition function of the dimer model. Any graph on a torus gives rise to a bipartite graph on the torus. We show that the phase space of the latter has a Lagrangian subvariety. We identify it with the space parametrizing resistor networks on the original graph.We construct several discrete quantum integrable systems.Comment: This is an updated version, 75 pages, which will appear in Ann. Sci. EN

Subclasses of Normal Helly Circular-Arc Graphs

A Helly circular-arc model M = (C,A) is a circle C together with a Helly family \A of arcs of C. If no arc is contained in any other, then M is a proper Helly circular-arc model, if every arc has the same length, then M is a unit Helly circular-arc model, and if there are no two arcs covering the circle, then M is a normal Helly circular-arc model. A Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc graph is the intersection graph of the arcs of a Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc model. In this article we study these subclasses of Helly circular-arc graphs. We show natural generalizations of several properties of (proper) interval graphs that hold for some of these Helly circular-arc subclasses. Next, we describe characterizations for the subclasses of Helly circular-arc graphs, including forbidden induced subgraphs characterizations. These characterizations lead to efficient algorithms for recognizing graphs within these classes. Finally, we show how do these classes of graphs relate with straight and round digraphs.Comment: 39 pages, 13 figures. A previous version of the paper (entitled Proper Helly Circular-Arc Graphs) appeared at WG'0