Vincent Limouzy

Vertex Intersection Graphs of Paths on a Grid

We investigate the class of vertex intersection graphs of paths on a grid, and specifically consider the subclasses that are obtained when each path in the representation has at most k bends (turns). We call such a subclass the Bk-VPG graphs, kC 0. In chip manufacturing, circuit layout is modeled as paths (wires) on a grid, where it is natural to constrain the number of bends per wire for reasons of feasibility and to reduce the cost of the chip. If the number k of bends is not restricted, then the VPG graphs are equivalent to the well-known class of string graphs, namely, the intersection graphs of arbitrary curves in the plane. In the case of B0-VPG graphs, we observe that horizontal and vertical segments have strong Helly number 2, and thus the clique problem has polynomial-time complexity, given the path representation. The recognition and coloring problems for B0-VPG graphs, however, are NPcomplete. We give a 2-approximation algorithm for coloring B0-VPG graphs. Furthermore, we prove that triangle-free B0-VPG graphs are 4-colorable, and this is best possible. We present a hierarchy of VPG graphs relating them to other known families of graphs. The grid intersection graphs are shown to be equivalent to the bipartite B0-VPG graphs and the circle graphs are strictly contained in B1-VPG. We prove the strict containment of B0-VPG into B1-VPG, and we conjecture that, in general, this strict containment continues for all values of k. We present a graph which is not in B1-VPG. Planar graphs are known to be in the class of string graphs, and we prove here that planar graphs are B3-VPG graphs, although it is not known if this is best possible.

String graphs of k-bend paths on a grid

Abstract We investigate the class of vertex intersection graphs of paths on a grid, and specifically consider the subclasses that are obtained when each path in the representation has at most k bends (turns). We call such a subclass the B k -VPG graphs, k ⩾ 0 . We present a complete hierarchy of VPG graphs relating them to other known families of graphs. String graphs are equivalent to VPG graphs. The grid intersection graphs [S. Bellantoni, I. Ben-Arroyo Hartman, T. Przytycka, S. Whitesides, Grid intersection graphs and boxicity, Discrete Math. 114, (1993), 41–49; I. Ben-Arroyo Hartman, I. Newman, R. Ziv, On grid intersection graphs, Discrete Math. 87(1), (1991), 41–52] are shown to be equivalent to the bipartite B 0 -VPG graphs. Chordal B 0 -VPG graphs are shown to be exactly Strongly Chordal B 0 -VPG graphs. We prove the strict containment of B 0 -VPG and circle graphs into B 1 -VPG. Planar graphs are known to be in the class of string graphs, and we prove here that planar graphs are B 3 -VPG graphs. In the case of B 0 -VPG graphs, we observe that a set of horizontal and vertical segments have strong Helly number 2. We show that the coloring problem for B k -VPG graphs, for k ⩾ 0 , is NP-complete and give a 2-approximation algorithm for coloring B 0 -VPG graphs. Furthermore, we prove that triangle-free B 0 -VPG graphs are 4-colorable, and this is best possible.

NLC-2 graph recognition and isomorphism

NLC-width is a variant of clique-width with many application in graph algorithmic. This paper is devoted to graphs of NLC-width two. After giving new structural properties of the class, we propose a O(n2m)-time algorithm, improving Johanssons algorithm [14]. Moreover, our alogrithm is simple to understand. The above properties and algorithm allow us to propose a robust O(n2m)-time isomorphism algorithm for NLC-2 graphs. As far as we know, it is the first polynomial-time algorithm.

Seidel Minor, Permutation Graphs and Combinatorial Properties

A permutation graph is an intersection graph of segments lying between two parallel lines. A Seidel complementation of a finite graph at a vertex v consists in complementing the edges between the neighborhood and the non-neighborhood of v. Two graphs are Seidel complement equivalent if one can be obtained from the other by a sequence of Seidel complementations.

Co-TT graphs and a characterization of split co-TT graphs

In this paper, we present a new characterization of complement Threshold Tolerance graphs (co-TT for short) and find a recognition algorithm for the subclass of split co-TT graphs running in O(n^2) time. Currently, the best recognition algorithms for co-TT graphs and for split co-TT graphs run in O(n^4) time (Hammer and Simeone (1981) [4]; Monma et al. (1988) [7]).

On some simplicial elimination schemes for chordal graphs

We present here some results on particular elimination schemes for chordal graphs, namely we show that for any chordal graph we can construct in linear time a simplicial elimination scheme starting with a pending maximal clique attached via a minimal separator maximal (resp. minimal) under inclusion among all minimal separators.

Unifying two graph decompositions with modular decomposition

We introduces the umodules, a generalization of the notion of graph module. The theory we develop captures among others undirected graphs, tournaments, digraphs, and 2-structures. We show that, under some axioms, a unique decomposition tree exists for umodules. Polynomial-time algorithms are provided for: non-trivial umodule test, maximal umodule computation, and decomposition tree computation when the tree exists. Our results unify many known decomposition like modular and bi-join decomposition of graphs, and a new decomposition of tournaments.

Hardness and Algorithms for Variants of Line Graphs of Directed Graphs

Given a directed graph D = (V,A) we define its intersection graph I(D) = (A,E) to be the graph having A as a node-set and two nodes of I(D) are adjacent if their corresponding arcs share a common node that is the tail of at least one of these arcs. We call them facility location graphs since they arise from the classical uncapacitated facility location problem. In this paper we show that facility location graphs are hard to recognize but they are easy to recognize when the underlying graph is triangle-free. We also determine the complexity of the vertex coloring, the stable set and the facility location problem for triangle-free facility location graphs.

On Modular Decomposition Concepts: the case for Homogeneous Relations

Modular decomposition has arisen in different contexts. In graph theory, it is fundamental. Indeed, many graph classes such as cographs or permutation graphs are characterised by specific properties on their modules. Moreover, well-known NP-complete problems such as colouring can be solved in polynomial (and often linear) time if the graph is “sufficiently” decomposable. A central point here relies on the decomposition theorem which presents a tree as compact encoding of the family of modules. Then, computing efficiently this tree given the graph has been an important challenge of the past two decades.

Bounds on Directed star arboricity in some digraph classes

Abstract A galaxy is a forest of directed stars. The notion of galaxy can be related to Facility Location problems as well as wavelength assignment problems in optical networks. Amini et al. [ Combinatorics, Probability & Computing , 19(2):161–182, 2010.] and Goncalves et al. [ Discrete Applied Mathematics , 160(6):744–754, 2012.] gave bounds on the minimum number of galaxies needed to cover the arcs of a digraph D , called directed star arboricity ( dst ( D )). They conjectured that those bounds could be improved such that dst ( D ) ≤ Δ ( D ) , for Δ ( D ) ≥ 3 and dst ( D ) ≤ 2 Δ + ( D ) for Δ + ( D ) ≥ 2 . In this work, we study the directed star arboricity in two non-trivial digraph classes: k -degenerate digraphs and tournaments.