Unit Interval Editing is Fixed-Parameter Tractable

Given a graph~$G$ and integers $k_1$, $k_2$, and~$k_3$, the unit interval editing problem asks whether $G$ can be transformed into a unit interval graph by at most $k_1$ vertex deletions, $k_2$ edge deletions, and $k_3$ edge additions. We give an algorithm solving this problem in time $2^{O(k\log k)}\cdot (n+m)$, where $k := k_1 + k_2 + k_3$, and $n, m$ denote respectively the numbers of vertices and edges of $G$. Therefore, it is fixed-parameter tractable parameterized by the total number of allowed operations. Our algorithm implies the fixed-parameter tractability of the unit interval edge deletion problem, for which we also present a more efficient algorithm running in time $O(4^k \cdot (n + m))$. Another result is an $O(6^k \cdot (n + m))$-time algorithm for the unit interval vertex deletion problem, significantly improving the algorithm of van 't Hof and Villanger, which runs in time $O(6^k \cdot n^6)$.Comment: An extended abstract of this paper has appeared in the proceedings of ICALP 2015. Update: The proof of Lemma 4.2 has been completely rewritten; an appendix is provided for a brief overview of related graph classe

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

On the structure of (pan, even hole)-free graphs

A hole is a chordless cycle with at least four vertices. A pan is a graph which consists of a hole and a single vertex with precisely one neighbor on the hole. An even hole is a hole with an even number of vertices. We prove that a (pan, even hole)-free graph can be decomposed by clique cutsets into essentially unit circular-arc graphs. This structure theorem is the basis of our $O(nm)$-time certifying algorithm for recognizing (pan, even hole)-free graphs and for our $O(n^{2.5}+nm)$-time algorithm to optimally color them. Using this structure theorem, we show that the tree-width of a (pan, even hole)-free graph is at most 1.5 times the clique number minus 1, and thus the chromatic number is at most 1.5 times the clique number.Comment: Accepted to appear in the Journal of Graph Theor

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

Linear-Time Algorithms for Finding Tucker Submatrices and Lekkerkerker-Boland Subgraphs

Lekkerkerker and Boland characterized the minimal forbidden induced subgraphs for the class of interval graphs. We give a linear-time algorithm to find one in any graph that is not an interval graph. Tucker characterized the minimal forbidden submatrices of binary matrices that do not have the consecutive-ones property. We give a linear-time algorithm to find one in any binary matrix that does not have the consecutive-ones property.Comment: A preliminary version of this work appeared in WG13: 39th International Workshop on Graph-Theoretic Concepts in Computer Scienc