How to construct equivalent differential systems

For two differential systems, an algorithm which permits us to say whether reflecting functions of these systems coincide or not is given. It allows studying qualitative properties of these two systems simultaneously.     

Connected graphs switching equivalent to their iterated line graphs

In this paper, the problem of determining graphs which are switching equivalent to at least one of their iterated line graphs is considered, and such connected graphs are characterized.     

An efficient algorithm to recognize locally equivalent graphs

To locally complement a simple graphF at one of its verticesv is to replace the subgraph induced byF onn(v)={w:vw is an edge ofF} by the complementary subgraph. Graphs related by a sequence of local complementations are said to be locally equivalent. We associate a system of equations with unknowns inGF(2) to any pair of graphs {F, F}, so thatF is locally equivalent toF if and only if the system has a solution. The equations are either linear and homogenous or bilinear, and we find a solution, if any, in polynomial time.With partial support of P. R. C. Mathématiques et Informatique.     

A note on the independence number of triangle-free graphs

Let G be a triangle-free graph on n points with average degree d. Let α be the independence number of G. In this note we give a simple proof that α ? n (d ln d ? d + 1)/(d ? 1)². We also consider what happens when G contains a limited number of triangles.     

Obstructions to chordal circular-arc graphs of small independence number

A blocking quadruple (BQ) is a quadruple of vertices of a graph such that any two vertices of the quadruple either miss (have no neighbours on) some path connecting the remaining two vertices of the quadruple, or are connected by some path missed by the remaining two vertices. This is akin to the notion of asteroidal triple used in the classical characterization of interval graphs by Lekkerkerker and Boland [Klee, V., What are the intersection graphs of arcs in a circle?, American Mathematical Monthly 76 (1976), pp. 810–813.].In this note, we first observe that blocking quadruples are obstructions for circular-arc graphs. We then focus on chordal graphs, and study the relationship between the structure of chordal graphs and the presence/absence of blocking quadruples.Our contribution is two-fold. Firstly, we provide a forbidden induced subgraph characterization of chordal graphs without blocking quadruples. In particular, we observe that all the forbidden subgraphs are variants of the subgraphs forbidden for interval graphs [Klee, V., What are the intersection graphs of arcs in a circle?, American Mathematical Monthly 76 (1976), pp. 810–813.]. Secondly, we show that the absence of blocking quadruples is sufficient to guarantee that a chordal graph with no independent set of size five is a circular-arc graph. In our proof we use a novel geometric approach, constructing a circular-arc representation by traversing around a carefully chosen clique tree.     

Acyclic directed graphs representing independence models

In this paper we study the problem of representing probabilistic independence models, in particular those closed under graphoid properties. We focus on acyclic directed graph (DAG): a new algorithm to build a DAG, given an ordering among random variables, is described and peculiarities and advantages of this approach are discussed. Moreover, we provide a necessary and sufficient condition for the existence of a perfect map representing an independence model and we describe an algorithm based on this characterization.     

Star clusters in independence complexes of graphs

We introduce the notion of star cluster of a simplex in a simplicial complex. This concept provides a general tool to study the topology of independence complexes of graphs. We use star clusters to answer a question arisen from works of Engström and Jonsson on the homotopy type of independence complexes of triangle-free graphs and to investigate a large number of examples which appear in the literature. We present an alternative way to study the chromatic and clique numbers of a graph from a homotopical point of view and obtain new results regarding the connectivity of independence complexes.     

Using data compressors to construct order tests for homogeneity and component independence

Nonparametric order tests for homogeneity and component independence are proposed, which are based on data compressors. For homogeneity testing the idea is to compress the word obtained by ordering the combined samples and writing the number of the sample in the place of each element. H₀ should be rejected if the string is compressed to a certain degree and accepted otherwise. We show that such a test obtained from an ideal data compressor is valid against all alternatives. Component independence is reduced to homogeneity testing.     

About equivalent interval colorings of weighted graphs

Given a graph G=(V,E) with strictly positive integer weights ω_i on the vertices iV, a k-interval coloring of G is a function I that assigns an interval I(i){1,…,k} of ω_i consecutive integers (called colors) to each vertex iV. If two adjacent vertices x and y have common colors, i.e. I(i)∩I(j)≠0/ for an edge [i,j] in G, then the edge [i,j] is said conflicting. A k-interval coloring without conflicting edges is said legal. The interval coloring problem (ICP) is to determine the smallest integer k, called interval chromatic number of G and denoted χ_int(G), such that there exists a legal k-interval coloring of G. For a fixed integer k, the k-interval graph coloring problem (k-ICP) is to determine a k-interval coloring of G with a minimum number of conflicting edges. The ICP and k-ICP generalize classical vertex coloring problems where a single color has to be assigned to each vertex (i.e., ω_i=1 for all vertices iV).Two k-interval colorings I₁ and I₂ are said equivalent if there is a permutation π of the integers 1,…,k such that ℓI₁(i) if and only if π(ℓ)I₂(i) for all vertices iV. As for classical vertex coloring, the efficiency of algorithms that solve the ICP or the k-ICP can be increased by avoiding considering equivalent k-interval colorings, assuming that they can be identified very quickly. To this purpose, we define and prove a necessary and sufficient condition for the equivalence of two k-interval colorings. We then show how a simple tabu search algorithm for the k-ICP can possibly be improved by forbidding the visit of equivalent solutions.     

On the independence number of random cubic graphs

We show that as n→∞, the independence number α(G), for almost all 3-regular graphs G on n vertices, is at least (6 log(3/2) – 2 – ?)n, for any constant ?>0. We prove this by analyzing a greedy algorithm for finding independent sets. © 1994 John Wiley & Sons, Inc.     

On the independence number of minimum distance graphs

Let F=F(n) denote the largest number such that any set of n points in the plane with minimum distance 1 has at least F elements, no two of which are at unit distance. Improving a result by Pollack, we show that F(n)≥ (9/35)n . We also give an O(n log n) time algorithm for selecting at least (9/35)n elements in a set of n points with minimum distance 1 so that no two selected points are at distance 1. <lsiheader> <onlinepub>7 August, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>20n2p179.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>no <sectionname> </lsiheader> Received June 12, 1996, and in revised form April 22, 1997.     

Large minors in graphs with given independence number

A weakening of Hadwiger’s conjecture states that every n-vertex graph with independence number α has a clique minor of size at least . Extending ideas of Fox (2010) [6], we prove that such a graph has a clique minor with at least vertices where c>1/19.2.     

Triangle-free graphs whose independence number equals the degree

In a triangle-free graph, the neighbourhood of every vertex is an independent set. We investigate the class S of triangle-free graphs where the neighbourhoods of vertices are maximum independent sets. Such a graph G must be regular of degree d=α(G) and the fractional chromatic number must satisfy χ_f(G)=|G|/α(G). We indicate that S is a rich family of graphs by determining the rational numbers c for which there is a graph G∈S with χ_f(G)=c except for a small gap, where we cannot prove the full statement. The statements for c≥3 are obtained by using, modifying, and re-analysing constructions of Sidorenko, Mycielski, and Bauer, van den Heuvel and Schmeichel, while the case c<3 is settled by a recent result of Brandt and Thomassé. We will also investigate the relation between other parameters of certain graphs in S like chromatic number and toughness.     

The independence polynomial of rooted products of graphs

A stable (or independent) set in a graph is a set of pairwise nonadjacent vertices thereof. The stability numberα(G) is the maximum size of stable sets in a graph G. The independence polynomial of G is

     

On clique covers and independence numbers of graphs

Relationships between the following graph invariants are studied: The node clique cover number, θ₀(>G); the clique cover number θ₁(G); node independence number, β₀(G); and an edge independence number, β′₁(G), We extend a theorem of Choudum, Parthasarathy and Ravindra with further statements equivalent to θ₀(G) = θ₁(G). More general results regarding the inequality θ₀(G) ? 0₁(G) are presented.     

A note on Vizing's independence number conjecture of edge chromatic critical graphs

In 1968, Vizing conjectured that, if G is a Δ-critical graph with n vertices, then α(G)?n/2, where α(G) is the independence number of G. In this note, we verify this conjecture for n?2Δ.     

Boolean techniques for matroidal decomposition of independence systems and applications to graphs

The problem of decomposing an independence system into the set-theoretic union of matroids is considered in the first part of this paper and a Boolean procedure is proposed for finding the prime matroidal components of such a decomposition. The second part of the paper deals with the special case of the independence system of all stable sets of a graph, characterizes the graphs whose family of stable sets is the set-theoretic union of two matroids, produces a class of perfect graphs of matroidal number k and gives for graphs an accelerated version of the general decomposition technique.     

Contributions to the theory of domination,independence and irredundance in graphs

A vertex x in a subset X of vertices of an undirected graph is redundant if its closed neighbourhood is contained in the union of closed neighbourhoods of vertices of X?{x}. In the context of a communications network, this means that any vertex which may receive communications from X may also be informed from X?{x}. The lower and upper irredundance numbers ir(G) and IR(G) are respectively the minimum and maximum cardinalities taken over all maximal sets of vertices having no redundancies. The domination number γ(G) and upper domination number Γ(G) are respectively the minimum and maximum cardinalities taken over all minimal dominating sets of G. The independent domination number i(G) and the independence number β(G) are respectively the minimum and maximum cardinalities taken over all maximal independent sets of vertices of G.A variety of inequalities involving these quantities are established and sufficient conditions for the equality of the three upper parameters are given. In particular a conjecture of Hoyler and Cockayne [9], namely $\text{i+β?2p + 2δ - 2}\text{2pδ}$, is proved.