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

Efficient enumeration of non-isomorphic interval graphs

Recently, Yamazaki et al. provided an algorithm that enumerates all non-isomorphic interval graphs on $n$ vertices with an $O(n^4)$ time delay. In this paper, we improve their algorithm and achieve $O(n^3 \log n)$ time delay. We also extend the catalog of these graphs providing a list of all non-isomorphic interval graphs for all $n$ up to $15$

Dynamic representation of consecutive-ones matrices and interval graphs

2015 Spring.Includes bibliographical references.We give an algorithm for updating a consecutive-ones ordering of a consecutive-ones matrix when a row or column is added or deleted. When the addition of the row or column would result in a matrix that does not have the consecutive-ones property, we return a well-known minimal forbidden submatrix for the consecutive-ones property, known as a Tucker submatrix, which serves as a certificate of correctness of the output in this case, in O(n log n) time. The ability to return such a certificate within this time bound is one of the new contributions of this work. Using this result, we obtain an O(n) algorithm for updating an interval model of an interval graph when an edge or vertex is added or deleted. This matches the bounds obtained by a previous dynamic interval-graph recognition algorithm due to Crespelle. We improve on Crespelle's result by producing an easy-to-check certificate, known as a Lekkerkerker-Boland subgraph, when a proposed change to the graph results in a graph that is not an interval graph. Our algorithm takes O(n log n) time to produce this certificate. The ability to return such a certificate within this time bound is the second main contribution of this work

Algebraic Operations on PQ Trees and Modular Decomposition Trees

Partitive set families are families of sets that can be quite large, but have a compact, recursive representation in the form of a tree. This tree is a common generalization of PQ trees, the modular decomposition of graphs, certain decompositions of boolean functions, and decompositions that arise on a variety of other combinatorial structures. We describe natural operators on partitive set families, give algebraic identities for manipulating them, and describe efficient algorithms for evaluating them. We use these results to obtain new time bounds for finding the common intervals of a set of permutations, finding the modular decomposition of an edge-colored graphs (also known as a two-structure), finding the PQ tree of a matrix when a consecutive-ones arrangement is given, and finding the modular decomposition of a permutation graph when its permutation realizer is given