On the sparsity order of a graph and its deficiency in chordality

Given a graph $G$ on $n$ nodes, let {cal P_G denote the cone consisting of the positive semidefinite $ntimes n$ matrices (with real or complex entries) having a zero entry at every position corresponding to a non edge of $G$. Then, the order of $G$ is defined as the maximum rank of a matrix lying on an extreme ray of the cone {cal P_G. It is shown in [AHMR88] that the graphs of order 1 are precisely the chordal graphs and a characterization of the graphs having order $2$ is conjectured there in the real case. We show in this paper the validity of this conjecture. Moreover, we characterize the graphs with order 2 in the complex case and we give a decomposition result for the graphs having order $le 2$ in both real and complex cases. As an application, these graphs can be recognized in polynomial time. We also establish an inequality relating the order {rm ord_{oF(G) of a graph $G$ ($oF=oR$ or $oC$) and the parameter {rm fill(G) defined as the minimum number of edges needed to be added to $G$ in order to obtain a chordal graph. Namely, we show that {rm ord_{oF(G)le 1 +epsilon_oF cdot {rm fill(G) where $epsilon_oR=1$ and $epsilon _oC =2$; this settles a conjecture posed in [HPR89]

A structure theorem for graphs with no cycle with a unique chord and its consequences

We give a structural description of the class C of graphs that do not contain a cycle with a unique chord as an induced subgraph. Our main theorem states that any connected graph in C is a either in some simple basic class or has a decomposition. Basic classes are cliques, bipartite graphs with one side containing only nodes of degree two and induced subgraph of the famous Heawood or Petersen graph. Decompositions are node cutsets consisting of one or two nodes and edge cutsets called 1-joins. Our decomposition theorem actually gives a complete structure theorem for C, i.e. every graph in C can be built from basic graphs that can be explicitly constructed, and gluing them together by prescribed composition operations ; and all graphs built this way are in C. This has several consequences : an O(nm)-time algorithm to decide whether a graph is in C, an O(n+m)-time algorithm that finds a maximum clique of any graph in C and an O(nm)-time coloring algorithm for graphs in C. We prove that every graph in C is either 3-colorable or has a coloring with ω colors where ω is the size of a largest clique. The problem of finding a maximum stable set for a graph in C is known to be NP-hard.Cycle with a unique chord, decomposition, structure, detection, recognition, Heawood graph, Petersen graph, coloring.

Graphs that do not contain a cycle with a node that has at least two neighbors on it

We recall several known results about minimally 2-connected graphs, and show that they all follow from a decomposition theorem. Starting from an analogy with critically 2-connected graphs, we give structural characterizations of the classes of graphs that do not contain as a subgraph and as an induced subgraph, a cycle with a node that has at least two neighbors on the cycle. From these characterizations we get polynomial time recognition algorithms for these classes, as well as polynomial time algorithms for vertex-coloring and edge-coloring

Extremal inhomogeneous Gibbs states for SOS-models and finite-spin models on trees

We consider $\mathbb Z$-valued $p$-SOS-models with nearest neighbor interactions of the form $|\omega_v-\omega_w|^p$, and finite-spin ferromagnetic models on regular trees. This includes the classical SOS-model, the discrete Gaussian model and the Potts model. We exhibit a family of extremal inhomogeneous (i.e. tree automorphism non-invariant) Gibbs measures arising as low temperature perturbations of ground states (local energy minimizers), which have a sparse enough set of broken bonds together with uniformly bounded increments along them. These low temperature states in general do not possess any symmetries of the tree. This generalises the results of Gandolfo, Ruiz and Shlosman \cite{GRS12} about the Ising model, and shows that the latter behaviour is robust. We treat three different types of extensions: non-compact state space gradient models, models without spin-symmetry, and models in small random fields. We give a detailed construction and full proofs of the extremality of the low-temperature states in the set of all Gibbs measures, analysing excess energies relative to the ground states, convergence of low-temperature expansions, and properties of cutsets.Comment: 27 pages, 3 figure

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

A structure theorem for graphs with no cycle with a unique chord and its consequences

URL des Documents de travail :http://ces.univ-paris1.fr/cesdp/CESFramDP2008.htmDocuments de travail du Centre d'Economie de la Sorbonne 2008.21 - ISSN : 1955-611XWe give a structural description of the class C of graphs that do not contain a cycle with a unique chord as an induced subgraph. Our main theorem states that any connected graph in C is a either in some simple basic class or has a decomposition. Basic classes are cliques, bipartite graphs with one side containing only nodes of degree two and induced subgraph of the famous Heawood or Petersen graph. Decompositions are node cutsets consisting of one or two nodes and edge cutsets called 1-joins. Our decomposition theorem actually gives a complete structure theorem for C, i.e. every graph in C can be built from basic graphs that can be explicitly constructed, and gluing them together by prescribed composition operations ; and all graphs built this way are in C. This has several consequences : an O(nm)-time algorithm to decide whether a graph is in C, an O(n+m)-time algorithm that finds a maximum clique of any graph in C and an O(nm)-time coloring algorithm for graphs in C. We prove that every graph in C is either 3-colorable or has a coloring with ω colors where ω is the size of a largest clique. The problem of finding a maximum stable set for a graph in C is known to be NP-hard.Soit C la classe de graphes ne contenant pas de cycle avec une seule corde en tant que sous-graphe induit. Nous montrons que tout graphe de C est ou bien "basique" ou bien "décomposable". Par graphe basique, nous entendons : clique, graphe biparti avec un côté ne contenant que des sommets de degré 2, et sous-graphe induit du graphe de Petersen ou de Heawood. Par décomposable, nous entendons : possédant un sommet ou une paire de sommets d'articulation ou possédant un 1-joint. Notre résultat est un théorème de structure, c'est-à-dire qu'il est réversible. Nous prouvons quelques conséquences : la preuve que tout graphe de C est ou bien coloriable avec 3 couleurs, ou bien avec ω couleurs où ω est la taille d'une plus grande clique. Nous donnons un algorithme de coloration en O(nm). Nous donnons un algorithme de reconnaissance en O(nm) pour la classe C. Cet algorithme répond à des questions intéressantes concernant la détection de sous-graphes induits

Vertex colouring and forbidden subgraphs - a survey

There is a great variety of colouring concepts and results in the literature. Here our focus is to survey results on vertex colourings of graphs defined in terms of forbidden induced subgraph conditions

Graph Theory

Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures

Polyhedral Problems in Combinatorial Convex Geometry

In this dissertation, we exhibit two instances of polyhedra in combinatorial convex geometry. The first instance arises in the context of Ehrhart theory, and the polyhedra are the central objects of study. The second instance arises in algebraic statistics, and the polyhedra act as a conduit through which we study a nonpolyhedral problem. In the first case, we examine combinatorial and algebraic properties of the Ehrhart h*-polynomial of the r-stable (n,k)-hypersimplices. These are a family of polytopes which form a nested chain of subpolytopes within the (n,k)-hypersimplex. We show that a well-studied unimodular triangulation of the (n,k)-hypersimplex restricts to a triangulation of each r-stable (n,k)-hypersimplex within. We then use this triangulation to compute the facet-defining inequalities of these polytopes. In the k=2 case, we use shelling techniques to devise a combinatorial interpretation of the coefficients of the h*-polynomials in terms of independent sets of certain graphs. From this, we then extract some results on unimodality. We also characterize the Gorenstein r-stable (n,k)-hypersimplices, and we conclude that these also have unimodal h*-polynomials. In the second case, for a graph G on p vertices we consider the closure of the cone of concentration matrices of G. The extreme rays of this cone, and their associated ranks, have applications in maximum likelihood estimation for the undirected Gaussian graphical model associated to G. Consequently, the extreme ranks of this cone have been well-studied. Yet, there are few graph classes for which all the possible extreme ranks are known. We show that the facet-normals of the cut polytope of G can serve to identify extreme rays of this nonpolyhedral cone. We see that for graphs without K5 minors each facet-normal of the cut polytope identifies an extreme ray in the cone, and we determine the rank of this extreme ray. When the graph is also series-parallel, we find that all possible extreme ranks arise in this fashion, thereby extending the collection of graph classes for which all the possible extreme ranks are known