Mixed unit interval graphs

AbstractThe class of intersection graphs of unit intervals of the real line whose ends may be open or closed is a strict superclass of the well-known class of unit interval graphs. We pose a conjecture concerning characterizations of such mixed unit interval graphs, verify parts of it in general, and prove it completely for diamond-free graphs. In particular, we characterize diamond-free mixed unit interval graphs by means of an infinite family of forbidden induced subgraphs, and we show that a diamond-free graph is mixed unit interval if and only if it has intersection representations using unit intervals such that all ends of the intervals are integral

A Lex-BFS-based recognition algorithm for Robinsonian matrices

Robinsonian matrices arise in the classical seriation problem and play an important role in many applications where unsorted similarity (or dissimilarity) information must be re- ordered. We present a new polynomial time algorithm to recognize Robinsonian matrices based on a new characterization of Robinsonian matrices in terms of straight enumerations of unit interval graphs. The algorithm is simple and is based essentially on lexicographic breadth-first search (Lex-BFS), using a divide-and-conquer strategy. When applied to a non- negative symmetric n × n matrix with m nonzero entries and given as a weighted adjacency list, it runs in O(d(n + m)) time, where d is the depth of the recursion tree, which is at most the number of distinct nonzero entries of A

Polynomial kernels for Proper Interval Completion and related problems

Given a graph G = (V,E) and a positive integer k, the Proper Interval Completion problem asks whether there exists a set F of at most k pairs of (V \times V)\E such that the graph H = (V,E \cup F) is a proper interval graph. The Proper Interval Completion problem finds applications in molecular biology and genomic research. First announced by Kaplan, Tarjan and Shamir in FOCS '94, this problem is known to be FPT, but no polynomial kernel was known to exist. We settle this question by proving that Proper Interval Completion admits a kernel with at most O(k^5) vertices. Moreover, we prove that a related problem, the so-called Bipartite Chain Deletion problem, admits a kernel with at most O(k^2) vertices, completing a previous result of Guo

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

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

A Fully Dynamic Algorithm for Recognizing and Representing Proper Interval Graphs

In this paper we study the problem of recognizing and representing dynamically changing proper interval graphs. The input to the problem consists of a series of modifications to be performed on a graph, where a modification can be a deletion or an addition of a vertex or an edge. The objective is to maintain a representation of the graph as long as it remains a proper interval graph, and to detect when it ceases to be so. The representation should enable one to efficiently construct a realization of the graph by an inclusion-free family of intervals. This problem has important applications in physical mapping of DNA

A Fully Dynamic Algorithm for Recognizing and Representing Proper Interval Graphs

In this paper we study the problem of recognizing and representing dynamically changing proper interval graphs. The input to the problem consists of a series of modifications to be performed on a graph, where a modification can be a deletion or an addition of a vertex or an edge. The objective is to maintain a representation of the graph as long as it remains a proper interval graph, and to detect when it ceases to be so. The representation should enable one to efficiently construct a realization of the graph by an inclusion-free family of intervals. This problem has important applications in physical mapping of DNA. We give a near-optimal fully dynamic algorithm for this problem. It operates in time O(log n) per edge insertion or deletion. We prove a close lower bound of\Omega\Gamma/24 n=(log log n + log b)) amortized time per operation, in the cell probe model with word-size b. We also construct optimal incremental and decremental algorithms for the problem, which handle each edge operation in O(1) time