The world of hereditary graph classes viewed through Truemper configurations

In 1982 Truemper gave a theorem that characterizes graphs whose edges can be labeled so that all chordless cycles have prescribed parities. The characterization states that this can be done for a graph G if and only if it can be done for all induced subgraphs of G that are of few speci c types, that we will call Truemper con gurations. Truemper was originally motivated by the problem of obtaining a co-NP characterization of bipartite graphs that are signable to be balanced (i.e. bipartite graphs whose node-node incidence matrices are balanceable matrices). The con gurations that Truemper identi ed in his theorem ended up playing a key role in understanding the structure of several seemingly diverse classes of objects, such as regular matroids, balanceable matrices and perfect graphs. In this survey we view all these classes, and more, through the excluded Truemper con gurations, focusing on the algorithmic consequences, trying to understand what structurally enables e cient recognition and optimization algorithms

Complexity of colouring problems restricted to unichord-free and \{square,unichord\}-free graphs

A \emph{unichord} in a graph is an edge that is the unique chord of a cycle. A \emph{square} is an induced cycle on four vertices. A graph is \emph{unichord-free} if none of its edges is a unichord. We give a slight restatement of a known structure theorem for unichord-free graphs and use it to show that, with the only exception of the complete graph $K_4$, every square-free, unichord-free graph of maximum degree~3 can be total-coloured with four colours. Our proof can be turned into a polynomial time algorithm that actually outputs the colouring. This settles the class of square-free, unichord-free graphs as a class for which edge-colouring is NP-complete but total-colouring is polynomial

Exact and Parameterized Algorithms for the Independent Cutset Problem

The Independent Cutset problem asks whether there is a set of vertices in a given graph that is both independent and a cutset. Such a problem is $\textsf{NP}$-complete even when the input graph is planar and has maximum degree five. In this paper, we first present a $\mathcal{O}^*(1.4423^{n})$-time algorithm for the problem. We also show how to compute a minimum independent cutset (if any) in the same running time. Since the property of having an independent cutset is MSO$_1$-expressible, our main results are concerned with structural parameterizations for the problem considering parameters that are not bounded by a function of the clique-width of the input. We present $\textsf{FPT}$-time algorithms for the problem considering the following parameters: the dual of the maximum degree, the dual of the solution size, the size of a dominating set (where a dominating set is given as an additional input), the size of an odd cycle transversal, the distance to chordal graphs, and the distance to $P_5$-free graphs. We close by introducing the notion of $\alpha$-domination, which allows us to identify more fixed-parameter tractable and polynomial-time solvable cases.Comment: 20 pages with references and appendi

The Strong Perfect Graph Conjecture: 40 years of Attempts, and its Resolution

International audienceThe Strong Perfect Graph Conjecture (SPGC) was certainly one of the most challenging conjectures in graph theory. During more than four decades, numerous attempts were made to solve it, by combinatorial methods, by linear algebraic methods, or by polyhedral methods. The first of these three approaches yielded the first (and to date only) proof of the SPGC; the other two remain promising to consider in attempting an alternative proof. This paper is an unbalanced survey of the attempts to solve the SPGC; unbalanced, because (1) we devote a signicant part of it to the 'primitive graphs and structural faults' paradigm which led to the Strong Perfect Graph Theorem (SPGT); (2) we briefly present the other "direct" attempts, that is, the ones for which results exist showing one (possible) way to the proof; (3) we ignore entirely the "indirect" approaches whose aim was to get more information about the properties and structure of perfect graphs, without a direct impact on the SPGC. Our aim in this paper is to trace the path that led to the proof of the SPGT as completely as possible. Of course, this implies large overlaps with the recent book on perfect graphs [J.L. Ramirez-Alfonsin and B.A. Reed, eds., Perfect Graphs (Wiley & Sons, 2001).], but it also implies a deeper analysis (with additional results) and another viewpoint on the topic

The complexity of the Perfect Matching-Cut problem

Perfect Matching-Cut is the problem of deciding whether a graph has a perfect matching that contains an edge-cut. We show that this problem is NP-complete for planar graphs with maximum degree four, for planar graphs with girth five, for bipartite five-regular graphs, for graphs of diameter three and for bipartite graphs of diameter four. We show that there exist polynomial time algorithms for the following classes of graphs: claw-free, $P_5$-free, diameter two, bipartite with diameter three and graphs with bounded tree-width

Minimal disconnected cuts in planar graphs

The problem of finding a disconnected cut in a graph is NP-hard in general but polynomial-time solvable on planar graphs. The problem of finding a minimal disconnected cut is also NP-hard but its computational complexity is not known for planar graphs. We show that it is polynomial-time solvable on 3-connected planar graphs but NP-hard for 2-connected planar graphs. Our technique for the first result is based on a structural characterization of minimal disconnected cuts in 3-connected K 3,3 -free-minor graphs and on solving a topological minor problem in the dual. We show that the latter problem can be solved in polynomial-time even on general graphs. In addition we show that the problem of finding a minimal connected cut of size at least 3 is NP-hard for 2-connected apex graphs

On Polynomial Kernelization of H\mathcal{H}-free Edge Deletion

For a set of graphs $\mathcal{H}$, the \textsc{$\mathcal{H}$-free Edge Deletion} problem asks to find whether there exist at most $k$ edges in the input graph whose deletion results in a graph without any induced copy of $H\in\mathcal{H}$. In \cite{cai1996fixed}, it is shown that the problem is fixed-parameter tractable if $\mathcal{H}$ is of finite cardinality. However, it is proved in \cite{cai2013incompressibility} that if $\mathcal{H}$ is a singleton set containing $H$, for a large class of $H$, there exists no polynomial kernel unless $coNP\subseteq NP/poly$. In this paper, we present a polynomial kernel for this problem for any fixed finite set $\mathcal{H}$ of connected graphs and when the input graphs are of bounded degree. We note that there are \textsc{$\mathcal{H}$-free Edge Deletion} problems which remain NP-complete even for the bounded degree input graphs, for example \textsc{Triangle-free Edge Deletion}\cite{brugmann2009generating} and \textsc{Custer Edge Deletion($P_3$-free Edge Deletion)}\cite{komusiewicz2011alternative}. When $\mathcal{H}$ contains $K_{1,s}$, we obtain a stronger result - a polynomial kernel for $K_t$-free input graphs (for any fixed $t> 2$). We note that for $s>9$, there is an incompressibility result for \textsc{$K_{1,s}$-free Edge Deletion} for general graphs \cite{cai2012polynomial}. Our result provides first polynomial kernels for \textsc{Claw-free Edge Deletion} and \textsc{Line Edge Deletion} for $K_t$-free input graphs which are NP-complete even for $K_4$-free graphs\cite{yannakakis1981edge} and were raised as open problems in \cite{cai2013incompressibility,open2013worker}.Comment: 12 pages. IPEC 2014 accepted pape