Absolute graphs with prescribed endomorphism monoid

We consider endomorphism monoids of graphs. It is well-known that any monoid can be represented as the endomorphism monoid M of some graph Γ with countably many colors. We give a new proof of this theorem such that the isomorphism between the endomorphism monoid $\mathop{\rm End}\nolimits (\Gamma)We consider endomorphism monoids of graphs. It is well-known that any monoid can be represented as the endomorphism monoid M of some graph Γ with countably many colors. We give a new proof of this theorem such that the isomorphism between the endomorphism monoid and M is absolute, i.e. holds in any generic extension of the given universe of set theory. This is true if and only if |M|,|Γ| are smaller than the first Erdős cardinal (which is known to be strongly inaccessible). We will encode Shelah’s absolutely rigid family of trees (Isr. J. Math. 42(3), 177–226, 1982) into Γ. The main result will be used to construct fields with prescribed absolute endomorphism monoids, see G?bel and Pokutta (Shelah’s absolutely rigid trees and absolutely rigid fields, in preparation). This work was supported by the project No. I-706-54.6/2001 of the German-Israeli Foundation for Scientific Research & Development and a fellowship within the Postdoc-Programme of the German Academic Exchange Service (DAAD).     

On s-hamiltonian line graphs of claw-free graphs

For an integer $s\ge 0$, a graph $G$ is $s$-hamiltonian if for any vertex subset $S?V\left(G\right)$ with $\left|S\right|\le s$, $G?S$ is hamiltonian, and $G$ is $s$-hamiltonian connected if for any vertex subset $S?V\left(G\right)$ with $\left|S\right|\le s$, $G?S$ is hamiltonian connected. Thomassen in 1984 conjectured that every 4-connected line graph is hamiltonian (see Thomassen, 1986), and Ku?zel and Xiong in 2004 conjectured that every 4-connected line graph is hamiltonian connected (see Ryjá?ek and Vrána, 2011). In Broersma and Veldman (1987), Broersma and Veldman raised the characterization problem of $s$-hamiltonian line graphs. In Lai and Shao (2013), it is conjectured that for $s\ge 2$, a line graph $L\left(G\right)$ is $s$-hamiltonian if and only if $L\left(G\right)$ is $(s+2)$-connected. In this paper we prove the following.(i) For an integer $s\ge 2$, the line graph $L\left(G\right)$ of a claw-free graph $G$ is $s$-hamiltonian if and only if $L\left(G\right)$ is $(s+2)$-connected.(ii) The line graph $L\left(G\right)$ of a claw-free graph $G$ is 1-hamiltonian connected if and only if $L\left(G\right)$ is 4-connected.     

Collapsible biclaw-free graphs

A graph is called biclaw-free if it has no biclaw as an induced subgraph. In this note, we prove that if G is a connected bipartite biclaw-free graph with δ(G)?5, then G is collapsible, and of course supereulerian. This bound is best possible.     

On distance two graphs of upper bound graphs

For a poset P=(X,≤), the upper bound graph (UB-graph) of P is the graph U=(X,E_U), where uv∈E_U if and only if u≠v and there exists m∈X such that u,v≤m. For a graph G, the distance two graph DS₂(G) is the graph with vertex set V(DS₂(G))=V(G) and u,v∈V(DS₂(G)) are adjacent if and only if d_G(u,v)=2. In this paper, we deal with distance two graphs of upper bound graphs. We obtain a characterization of distance two graphs of split upper bound graphs.     

Partial characterizations of clique-perfect graphs I: Subclasses of claw-free graphs

A clique-transversal of a graph G is a subset of vertices that meets all the cliques of G. A clique-independent set is a collection of pairwise vertex-disjoint cliques. The clique-transversal number and clique-independence number of G are the sizes of a minimum clique-transversal and a maximum clique-independent set of G, respectively. A graph G is clique-perfect if these two numbers are equal for every induced subgraph of G. The list of minimal forbidden induced subgraphs for the class of clique-perfect graphs is not known. In this paper, we present a partial result in this direction; that is, we characterize clique-perfect graphs by a restricted list of forbidden induced subgraphs when the graph belongs to two different subclasses of claw-free graphs.     

On the Zagreb indices of the line graphs of the subdivision graphs

The aim of this paper is to investigate the Zagreb indices of the line graphs of the tadpole graphs, wheel graphs and ladder graphs using the subdivision concepts.     

Partial characterizations of clique-perfect graphs II: Diamond-free and Helly circular-arc graphs

A clique-transversal of a graph G is a subset of vertices that meets all the cliques of G. A clique-independent set is a collection of pairwise vertex-disjoint cliques. A graph G is clique-perfect if the sizes of a minimum clique-transversal and a maximum clique-independent set are equal for every induced subgraph of G. The list of minimal forbidden induced subgraphs for the class of clique-perfect graphs is not known. Another open question concerning clique-perfect graphs is the complexity of the recognition problem. Recently we were able to characterize clique-perfect graphs by a restricted list of forbidden induced subgraphs when the graph belongs to two different subclasses of claw-free graphs. These characterizations lead to polynomial time recognition of clique-perfect graphs in these classes of graphs. In this paper we solve the characterization problem in two new classes of graphs: diamond-free and Helly circular-arc () graphs. This last characterization leads to a polynomial time recognition algorithm for clique-perfect graphs.     

Equistable distance-hereditary graphs

A graph is called equistable when there is a non-negative weight function on its vertices such that a set S of vertices has total weight 1 if and only if S is maximal stable. We show that a necessary condition for a graph to be equistable is sufficient when the graph in question is distance-hereditary. This is used to design a polynomial-time recognition algorithm for equistable distance-hereditary graphs.     

Marginal AMP chain graphs

We present a new family of models that is based on graphs that may have undirected, directed and bidirected edges. We name these new models marginal AMP (MAMP) chain graphs because each of them is Markov equivalent to some AMP chain graph under marginalization of some of its nodes. However, MAMP chain graphs do not only subsume AMP chain graphs but also multivariate regression chain graphs. We describe global and pairwise Markov properties for MAMP chain graphs and prove their equivalence for compositional graphoids. We also characterize when two MAMP chain graphs are Markov equivalent.For Gaussian probability distributions, we also show that every MAMP chain graph is Markov equivalent to some directed and acyclic graph with deterministic nodes under marginalization and conditioning on some of its nodes. This is important because it implies that the independence model represented by a MAMP chain graph can be accounted for by some data generating process that is partially observed and has selection bias. Finally, we modify MAMP chain graphs so that they are closed under marginalization for Gaussian probability distributions. This is a desirable feature because it guarantees parsimonious models under marginalization.     

Two-ended regular median graphs

We show that regular median graphs of linear growth are the Cartesian product of finite hypercubes with the two-way infinite path. Such graphs are Cayley graphs and have only two ends.For cubic median graphs G the condition of linear growth can be weakened to the condition that G has two ends. For higher degree the relaxation to two-ended graphs is not possible, which we demonstrate by an example of a median graph of degree four that has two ends, but nonlinear growth.     

Boxicity of Halin graphs

A k-dimensional box is the Cartesian product R₁×R₂×?×R_k where each R_i is a closed interval on the real line. The boxicity of a graph G, denoted as is the minimum integer k such that G is the intersection graph of a collection of k-dimensional boxes. Halin graphs are the graphs formed by taking a tree with no degree 2 vertex and then connecting its leaves to form a cycle in such a way that the graph has a planar embedding. We prove that if G is a Halin graph that is not isomorphic to K₄, then . In fact, we prove the stronger result that if G is a planar graph formed by connecting the leaves of any tree in a simple cycle, then unless G is isomorphic to K₄ (in which case its boxicity is 1).     

On conservative and supermagic graphs

A graph is called supermagic if it admits a labelling of the edges by pairwise different consecutive positive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. A graph G is called conservative if it admits an orientation and a labelling of the edges by integers {1,…,|E(G)|} such that at each vertex the sum of the labels on the incoming edges is equal to the sum of the labels on the outgoing edges. In this paper we deal with conservative graphs and their connection with the supermagic graphs. We introduce a new method to construct supermagic graphs using conservative graphs. Inter alia we show that the union of some circulant graphs and regular complete multipartite graphs are supermagic.     

Self-clique Helly circular-arc graphs

A clique in a graph is a complete subgraph maximal under inclusion. The clique graph of a graph is the intersection graph of its cliques. A graph is self-clique when it is isomorphic to its clique graph. A circular-arc graph is the intersection graph of a family of arcs of a circle. A Helly circular-arc graph is a circular-arc graph admitting a model whose arcs satisfy the Helly property. In this note, we describe all the self-clique Helly circular-arc graphs.     

Line-graphs of cubic graphs are normal

A graph is called normal if its vertex set can be covered by cliques Q₁,Q₂,…,Q_k and also by stable sets S₁,S₂,…,S_l, such that S_i∩Q_j≠∅ for every i,j. This notion is due to Körner, who introduced the class of normal graphs as an extension of the class of perfect graphs. Normality has also relevance in information theory. Here we prove, that the line graphs of cubic graphs are normal.     

Stochastic kronecker graphs

A random graph model based on Kronecker products of probability matrices has been recently proposed as a generative model for large‐scale real‐world networks such as the web. This model simultaneously captures several well‐known properties of real‐world networks; in particular, it gives rise to a heavy‐tailed degree distribution, has a low diameter, and obeys the densification power law. Most properties of Kronecker products of graphs (such as connectivity and diameter) are only rigorously analyzed in the deterministic case. In this article, we study the basic properties of stochastic Kronecker products based on an initiator matrix of size two (which is the case that is shown to provide the best fit to many real‐world networks). We will show a phase transition for the emergence of the giant component and another phase transition for connectivity, and prove that such graphs have constant diameters beyond the connectivity threshold, but are not searchable using a decentralized algorithm. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 453–466, 2011     

Recognizing Helly Edge-Path-Tree graphs and their clique graphs

We present a unifying procedure for recognizing intersection graphs of Helly families of paths in a tree and their clique graphs. The Helly property makes it possible to look at these recognition problems as variants of the Graph Realization Problem, namely, the problem of recognizing Edge-Path-Tree matrices. Our result heavily relies on the notion of pie introduced in [M.C. Golumbic, R.E. Jamison, The edge intersection graphs of paths in a tree, Journal of Combinatorial Theory, Series B 38 (1985) 8-22] and on the observation that Helly Edge-Path-Tree matrices form a self-dual class of Helly matrices. Coupled to the notion of reduction presented in the paper, these facts are also exploited to reprove and slightly refine some known results for Edge-Path-Tree graphs.     

Partial characterizations of coordinated graphs: line graphs and complements of forests

A graph G is coordinated if the minimum number of colors that can be assigned to the cliques of H in such a way that no two cliques with non-empty intersection receive the same color is equal to the maximum number of cliques of H with a common vertex, for every induced subgraph H of G. Coordinated graphs are a subclass of perfect graphs. The list of minimal forbidden induced subgraphs for the class of coordinated graphs is not known. In this paper, we present a partial result in this direction, that is, we characterize coordinated graphs by minimal forbidden induced subgraphs when the graph is either a line graph, or the complement of a forest. F. Bonomo, F. Soulignac, and G. Sueiro’s research partially supported by UBACyT Grant X184 (Argentina), and CNPq under PROSUL project Proc. 490333/2004-4 (Brazil). The research of G. Durán is partially supported by FONDECyT Grant 1080286 and Millennium Science Institute “Complex Engineering Systems” (Chile), and CNPq under PROSUL project Proc. 490333/2004-4 (Brazil).