Given a graph $G$, a 2-coloring of the edges of $K_n$ is said to contain a balanced copy of $G$ if we can find a copy of $G$ such that half of its edges are in each color class. If, for every sufficiently large $n$, there exists an integer $k$ such that every 2-coloring of $K_n$ with more than $k$ edges in each color class contains a balanced copy of $G$, then we say that $G$ is balanceable. Balanceability was introduced by Caro, Hansberg and Montejano, who also gave a structural characterization of balanceable graphs.
In this paper, we extend the study of balanceability by finding new sufficient conditions for a graph to be balanceable or not. We use those conditions to fully characterize the balanceability of graph classes such as rectangular and triangular grids, as well as a special class of circulant graphs.
We investigate the paramater of the average range of $M$-Lipschitz mapping of a given graph. We focus on well-known classes such as paths, complete graphs, complete bipartite graphs and cycles and show closed formulas for computing this parameter and also we conclude asymptotics of this parameter on these aforementioned classes.
Baker devised a powerful technique to obtain approximation schemes for various problems restricted to planar graphs. Her technique can be directly extended to various other graph classes, among the most general ones the graphs avoiding a fixed apex graph as a minor. Further generalizations (e.g., to all proper minor closed graph classes) are known, but they use a combination of techniques and usually focus on somewhat restricted classes of problems. We present a new type of graph decompositions (thin systems of overlays) generalizing Baker's technique and leading to straightforward polynomial-…
The infinite tumbling block graph is a bipartite graph, where each vertex in one partite set is of degree 3 and each vertex in the other partite set is of degree 6. It is a 2-dimensional array of blocks of seven vertices and nine edges, a planar graph that has 3-D looks. This paper introduces tumbling block graphs and considers various graphical parameters for different classes of infinite and finite tumbling blocks.
Nowhere dense graph classes provide one of the least restrictive notions of sparsity for graphs. Several equivalent characterizations of nowhere dense classes have been obtained over the years, using a wide range of combinatorial objects. In this paper we establish a new characterization of nowhere dense classes, in terms of poset dimension: A monotone graph class is nowhere dense if and only if for every $h \geq 1$ and every $ε> 0$, posets of height at most $h$ with $n$ elements and whose cover graphs are in the class have dimension $\mathcal{O}(n^ε)$.
We provide a general method to prove the existence and compute efficiently elimination orderings in graphs. Our method relies on several tools that were known before, but that were not put together so far: the algorithm LexBFS due to Rose, Tarjan and Lueker, one of its properties discovered by Berry and Bordat, and a local decomposition property of graphs discovered by Maffray, Trotignon and Vu\vsković. We use this method to prove the existence of elimination orderings in several classes of graphs, and to compute them in linear time. Some of the classes have already been studied, namely even-h…
A graph is an opposition graph, respectively, a coalition graph, if it admits an acyclic orientation which puts the two end-edges of every chordless 4-vertex path in opposition, respectively, in the same direction. Opposition and coalition graphs have been introduced and investigated in connection to perfect graphs. Recognizing and characterizing opposition and coalition graphs are long-standing open problems. This paper gives characterizations for opposition graphs and coalition graphs on some restricted graph classes. Implicit in our arguments are polynomial time recognition algorithms for t…
In this paper, we introduce the notion of the containment graph of a family of sets and containment classes of graphs and posets. Let $Z$ be a family of nonempty sets. We call a (simple, finite) graph G = (V, E) a $Z$-containment graph provided one can assign to each vertex $v_i \in V $ a set $S_i \in Z$ such that $v_i v_j \in E$ if and only if $S_i \subset S_j$ or $S_j \subset S_i$ . Similarly, we call a (strict) partially ordered set $P = (V, <)$ a $Z$-containment poset if to each $v_i \in V $ we can assign a set $S_i \in Z$ such that $v_i < v_j$ if and only if $S_i \subset S_j$. Obviously, …
For a set $S$ of vertices and the vertex $v$ in a connected graph $G$, $\displaystyle\max_{x \in S}d(x,v)$ is called the $S$-eccentricity of $v$ in $G$. The set of vertices with minimum $S$-eccentricity is called the $S$-center of $G$. Any set $A$ of vertices of $G$ such that $A$ is an $S$-center for some set $S$ of vertices of $G$ is called a center set. We identify the center sets of certain classes of graphs namely, Block graphs, $K_{m,n}$, $K_n-e$, wheel graphs, odd cycles and symmetric even graphs and enumerate them for many of these graph classes. We also introduce the concept of center …
The celebrated theorem of Robertson and Seymour states that in the family of minor-closed graph classes, there is a unique minimal class of graphs of unbounded tree-width, namely, the class of planar graphs. In the case of tree-width, the restriction to minor-closed classes is justified by the fact that the tree-width of a graph is never smaller than the tree-width of any of its minors. This, however, is not the case with respect to clique-width, as the clique-width of a graph can be (much) smaller than the clique-width of its minor. On the other hand, the clique-width of a graph is never smal…
This paper is the first from serial papers that provide constructive characterizations for classes of bidirected graphs known as radials and semiradials. In this paper, we provide constructive characterizations for five principle classes of radials and semiradials to be used for characterizing general radials and semiradials. A bidirected graph is a graph in which each end of each edge has a sign $+$ or $-$. Bidirected graphs are a common generalization of digraphs and signed graphs. We define a new concept of radials as a generalization of a classical concept in matching theory, critical grap…
The notion of nowhere denseness is one of the central concepts of the recently developed theory of sparse graphs. We study the properties of nowhere dense graph classes by investigating appropriate limit objects defined using the ultraproduct construction. It appears that different equivalent definitions of nowhere denseness, for example via quasi-wideness or the splitter game, correspond to natural notions for the limit objects that are conceptually simpler and allow for less technically involved reasonings.
A proper vertex coloring of a graph is said to be locally identifying if the sets of colors in the closed neighborhood of any two adjacent non-twin vertices are distinct. The lid-chromatic number of a graph is the minimum number of colors used by a locally identifying vertex-coloring. In this paper, we prove that for any graph class of bounded expansion, the lid-chromatic number is bounded. Classes of bounded expansion include minor closed classes of graphs. For these latter classes, we give an alternative proof to show that the lid-chromatic number is bounded. This leads to an explicit upper …
In this paper, we reveal an intriguing relationship between two seemingly unrelated notions: letter graphs and geometric grid classes of permutations. An important property common for both of them is well-quasi-orderability, implying, in a non-constructive way, a polynomial-time recognition of geometric grid classes of permutations and $k$-letter graphs for a fixed $k$. However, constructive algorithms are available only for $k=2$. In this paper, we present the first constructive polynomial-time algorithm for the recognition of $3$-letter graphs. It is based on a structural characterization of…
A graph is Helly if its disks satisfy the Helly property, i.e., every family of pairwise intersecting disks in G has a common intersection. It is known that for every graph G, there exists a unique smallest Helly graph H(G) into which G isometrically embeds; H(G) is called the injective hull of G. Motivated by this, we investigate the structural properties of the injective hulls of various graph classes. We say that a class of graphs $\mathcal{C}$ is closed under Hellification if $G \in \mathcal{C}$ implies $H(G) \in \mathcal{C}$. We identify several graph classes that are closed under Hellifi…
It was noted already in the 90s that many classic graph classes, such as interval, chordal, and bipartite graphs, can be characterized by the existence of an ordering of the vertices avoiding some ordered subgraphs, called patterns. Very recently, all the classes corresponding to patterns on three vertices (including the ones mentioned above) have been listed, and proved to be efficiently recognizable. In contrast, very little is known about patterns on four vertices. One of the few graph classes characterized by a pattern on four vertices is the class of intersection graphs of rectangles that…
A graph is called odd (respectively, even) if every vertex has odd (respectively, even) degree. Gallai proved that every graph can be partitioned into two even induced subgraphs, or into an odd and an even induced subgraph. We refer to a partition into odd subgraphs as an odd colouring of G. Scott [Graphs and Combinatorics, 2001] proved that a graph admits an odd colouring if and only if it has an even number of vertices. We say that a graph G is k-odd colourable if it can be partitioned into at most k odd induced subgraphs. We initiate the systematic study of odd colouring and odd chromatic n…
We discover new hereditary classes of graphs that are minimal (with respect to set inclusion) of unbounded clique-width. The new examples include split permutation graphs and bichain graphs. Each of these classes is characterised by a finite list of minimal forbidden induced subgraphs. These, therefore, disprove a conjecture due to Daligault, Rao and Thomasse from 2010 claiming that all such minimal classes must be defined by infinitely many forbidden induced subgraphs. In the same paper, Daligault, Rao and Thomasse make another conjecture that every hereditary class of unbounded clique-width …
If a graph has no induced subgraph isomorphic to any graph in a finite family $\{H_1,\ldots,H_p\}$, it is said to be $(H_1,\ldots,H_p)$-free. The class of $H$-free graphs has bounded clique-width if and only if $H$ is an induced subgraph of the 4-vertex path $P_4$. We study the (un)boundedness of the clique-width of graph classes defined by two forbidden induced subgraphs $H_1$ and $H_2$. Prior to our study it was not known whether the number of open cases was finite. We provide a positive answer to this question. To reduce the number of open cases we determine new graph classes of bounded cli…
An identifying code of a graph is a subset of its vertices such that every vertex of the graph is uniquely identified by the set of its neighbours within the code. We show a dichotomy for the size of the smallest identifying code in classes of graphs closed under induced subgraphs. Our dichotomy is derived from the VC-dimension of the considered class C, that is the maximum VC-dimension over the hypergraphs formed by the closed neighbourhoods of elements of C. We show that hereditary classes with infinite VC-dimension have infinitely many graphs with an identifying code of size logarithmic in …
Motivated by Chudnovsky's structure theorem of bull-free graphs, Abu-Khzam, Feghali, and Müller have recently proved that deciding if a graph has a vertex partition into disjoint cliques and a triangle-free graph is NP-complete for five graph classes. The problem is trivial for the intersection of these five classes. We prove that the problem is NP-complete for the intersection of two subsets of size four among the five classes. We also show NP-completeness for other small classes, such as graphs with maximum degree 4 and line graphs.
We prove that a minor-closed class of graphs has bounded layered pathwidth if and only if some apex-forest is not in the class. This generalises a theorem of Robertson and Seymour, which says that a minor-closed class of graphs has bounded pathwidth if and only if some forest is not in the class.
Graph embedding is the major technique which is used to map guest graph into host graph. In architecture simulation, graph embedding is said to be one of the strongest application for the execution of parallel algorithm and simulation of various interconnection networks \cite{Pa99}. In this paper, we have embedded circulant networks into star of cycle and folded hypercube into cycle-of-ladders and compute its exact wirelength. Further we have discussed the embedding parameters in VLSI Layout.
We introduce the concept of shallow directed minors and based on this a new classification of classes of directed graphs which is diametric to existing directed graph decompositions and width measures proposed in the literature. We then study in depth one type of classes of directed graphs which we call nowhere crownful. The classes are very general as they include, on one hand, all classes of directed graphs whose underlying undirected class is nowhere dense, such as planar, bounded-genus, and $H$-minor-free graphs; and on the other hand, also contain classes of high edge density whose underl…
In this paper, we have obtained the total chromatic as well as equitable and neighborhood sum distinguishing total chromatic numbers of some classes of the circulant graphs.
We study the first-order (FO) model checking problem of dense graphs, namely those which have FO interpretations in (or are FO transductions of) some sparse graph classes. We give a structural characterization of the graph classes which are FO interpretable in graphs of bounded degree. This characterization allows us to efficiently compute such an FO interpretation for an input graph. As a consequence, we obtain an FPT algorithm for successor-invariant FO model checking of any graph class which is FO interpretable in (or an FO transduction of) a graph class of bounded degree. The approach we u…
A hypergraph is said to be $1$-Sperner if for every two hyperedges the smallest of their two set differences is of size one. We present several applications of $1$-Sperner hypergraphs and their structure to graphs. In particular, we consider the classical characterizations of threshold and domishold graphs and use them to obtain further characterizations of these classes in terms of $1$-Spernerness, thresholdness, and $2$-asummability of their vertex cover, clique, dominating set, and closed neighborhood hypergraphs. Furthermore, we apply a decomposition property of $1$-Sperner hypergraphs to …
This work provides a structural characterisation of hereditary graph classes that do not contain a star forest, several graphs obtained from star forests by subset complementation, a union of cliques, and the complement of a union of cliques as induced subgraphs. This provides, for instance, structural results for graph classes not containing a matching and several complements of a matching. In terms of the speed of hereditary graph classes, our results imply that all such classes have at most factorial speed of growth.
This paper deals with graph classes characterization and recognition. A popular way to characterize a graph class is to list a minimal set of forbidden induced subgraphs. Unfortunately this strategy usually does not lead to an efficient recognition algorithm. On the other hand, many graph classes can be efficiently recognized by techniques based on some interesting orderings of the nodes, such as the ones given by traversals. We study specifically graph classes that have an ordering avoiding some ordered structures. More precisely, we consider what we call patterns on three nodes, and the reco…
We construct minor-closed addable families of graphs that are subcritical and contain all planar graphs. This contradicts (one direction of) a well-known conjecture of Noy.
Classes with bounded rankwidth are MSO-transductions of trees and classes with bounded linear rankwidth are MSO-transductions of paths -- a result that shows a strong link between the properties of these graph classes considered from the point of view of structural graph theory and from the point of view of finite model theory. We take both views on classes with bounded linear rankwidth and prove structural and model theoretic properties of these classes. The structural results we obtain are the following. 1) The number of unlabeled graphs of order $n$ with linear rank-width at most~$r$ is at …
It was recently proved that every planar graph is a subgraph of the strong product of a path and a graph with bounded treewidth. This paper surveys generalisations of this result for graphs on surfaces, minor-closed classes, various non-minor-closed classes, and graph classes with polynomial growth. We then explore how graph product structure might be applicable to more broadly defined graph classes. In particular, we characterise when a graph class defined by a cartesian or strong product has bounded or polynomial expansion. We then explore graph product structure theorems for various geometr…
We consider several graphs classes defined in terms of conditions on cliques and stable sets, including CIS, split, equistable, and other related classes. We pursue a systematic study of the relations between them. As part of this study, we introduce two generalizations of CIS graphs, obtain a new characterization of split graphs, and a characterization of CIS line graphs.
The processes of constructing some graphs from others using binary operations of union with intersection (gluing) are studied. For graph classes closed with respect to gluing operations the elemental and operational bases are introduced. The generating bases together with the system of restrictions on the gluing operation that preserves the characteristic properties of graphs form constructive descriptions of the closed classes of graphs. It is shown that each closed class of graphs has a unique elemental basis and at least one operational basis. A constructive description of a closed class of…
The article considers the procedure of connection of graphs to the edges of a cyclic graph and its influence on the sandpile group of the graph thus obtained. A series of classes of graphs CH_n(a_1,...,a_n) is defined. Recurrent and non-recurrent formulas for calculating the sandpile groups of all graphs of classes CH_n (a_1, ..., a_n) are proposed.
Minimal separators in graphs are an important concept in algorithmic graph theory. In particular, many problems that are NP-hard for general graphs are known to become polynomial-time solvable for classes of graphs with a polynomially bounded number of minimal separators. Several well-known graph classes have this property, including chordal graphs, permutation graphs, circular-arc graphs, and circle graphs. We perform a systematic study of the question which classes of graphs defined by small forbidden induced subgraphs have a polynomially bounded number of minimal separators. We focus on set…
Frame matroids and lifted-graphic matroids are two distinct minor-closed classes of matroids, each of which generalises the class of graphic matroids. The class of quasi-graphic matroids, recently introduced by Geelen, Gerards, and Whittle, simultaneously generalises both the classes of frame and lifted-graphic matroids. Let $\mathcal{M}$ be one of these three classes, and let $r$ be a positive integer. We show that $\mathcal{M}$ has only a finite number of excluded minors of rank $r$.
In this paper, we construct intriguing sets in five classes of strongly regular graphs defined on nonisotropic points of finite classical polar spaces, and determine their intersection numbers.
A rainbow graph is a graph that admits a vertex-coloring such that every color appears exactly once in the neighborhood of each vertex. We investigate some properties of rainbow graphs. In particular, we show that there is a bijection between the isomorphism classes of n-rainbow graphs on 2n vertices and the switching classes of graphs on n vertices.
Two edge colorings of a graph are {\em edge-Kempe equivalent} if one can be obtained from the other by a series of edge-Kempe switches. This work gives some results for the number of edge-Kempe equivalence classes for cubic graphs. In particular we show every 2-connected planar bipartite cubic graph has exactly one edge-Kempe equivalence class. Additionally, we exhibit infinite families of nonplanar bipartite cubic graphs with a range of numbers of edge-Kempe equivalence classes. Techniques are developed that will be useful for analyzing other classes of graphs as well.
The classes of odd graphs $O_n$ and middle levels graphs $B_n$ form one parameter subclasses of the Kneser graphs and bipartite Kneser graphs respectively. In particular both classes are vertex transitive while resisting definitive conclusions about their Hamiltonicity, and have thus come under scrutiny with regards to the Lovász conjecture. In this paper, we will establish that middle levels graphs may always be embedded in odd graphs and middle levels graphs of higher degree, and furthermore, that this embedding allows us to define a recursion relationship in both classes which can be used t…
We study the complexity of the Graph Isomorphism problem on graph classes that are characterized by a finite number of forbidden induced subgraphs, focusing mostly on the case of two forbidden subgraphs. We show hardness results and develop techniques for the structural analysis of such graph classes, which applied to the case of two forbidden subgraphs give the following results: A dichotomy into isomorphism complete and polynomial-time solvable graph classes for all but finitely many cases, whenever neither of the forbidden graphs is a clique, a pan, or a complement of these graphs. Further …
We enumerate the independent sets of several classes of regular and almost regular graphs and compute the corresponding generating functions. We also note the relations between these graphs and other combinatorial objects and, in some cases, construct the corresponding bijections.
We give two constructions of strongly regular Cayley graphs on finite fields $\F_q$ by using union of cyclotomic classes and index 2 Gauss sums. In particular, we obtain twelve infinite families of strongly regular graphs with new parameters.
A $k$-regular graph is called a divisible design graph if its vertex set can be partitioned into $m$ classes of size $n$, such that two distinct vertices from the same class have exactly $λ_1$ common neighbours, and two vertices from different classes have exactly $λ_2$ common neighbours. In this paper, we find the vertex connectivity of some classes of divisible design graphs, in particular, we present examples of divisible design graphs, whose vertex connectivity is less than $k$, where $k$ is the degree of a vertex. We also show that the vertex connectivity a divisible design graphs may be …
This document is the Habilitation Thesis (Habilitation à Diriger des Recherches) of Nicolas Trotignon. It presents his work in structural graph theory, from 2001 to 2009.
The bounds on the sum and product of chromatic numbers of a graph and its complement are known as Nordhaus-Gaddum inequalities. In this paper, we study the operations on the Independence numbers of graphs with their complement. We also provide a new characterization of certain graph classes.
For a graph $G$, let $τ(G)$ be the maximum number of colors such that there exists an edge-coloring of $G$ with no two color classes being isomorphic. We investigate the behavior of $τ(G)$ when $G=G(n, p)$ is the classical Erdős-Rényi random graph.
A graph $U$ is universal for a graph class $\mathcal{C}\ni U$, if every $G\in \mathcal{C}$ is a minor of $U$. We prove the existence or absence of universal graphs in several natural graph classes, including graphs component-wise embeddable into a surface, and graphs forbidding $K_5$, or $K_{3,3}$, or $K_\infty$ as a minor. We prove the existence of uncountably many minor-closed classes of countable graphs that (do and) do not have a universal element. Some of our results and questions may be of interest to the finite graph theorist. In particular, one of our side-results is that every $K_5$-m…
