Topics in Discrete Mathematics (special edition for J. Nesetril, eds. M. Klazar, J. Kratochvil, M. Loebl, J. Matousek, R. Thomas and P. Valtr), Springer, 26 (2006), 179-213
An ordered graph is a graph together with a linear order on its vertices. A hereditary property of ordered graphs is a collection of ordered graphs closed under taking induced ordered subgraphs. If P is a property of ordered graphs, then the function which counts the number of ordered graphs in P with exactly n vertices is called the speed of P.
In this paper we determine the possible speeds of a hereditary property of ordered graphs, up to the speed 2^(n-1). In particular, we prove that there exists a jump from polynomial speed to speed F(n), the Fibonacci numbers, and that there exists an infinite sequence of subsequent jumps, from p(n)F(n,k) to F(n,k+1) (where p(n) is a polynomial and F(n,k) are the generalized Fibonacci numbers) converging to 2^(n-1). Our results generalize a theorem of Kaiser and Klazar, who proved that the same jumps occur for hereditary properties of permutations.
In this paper we use the Klazar-Marcus-Tardos method to prove that if a hereditary property of partitions P has super-exponential speed, then for every k-permutation pi, P contains the partition of [2k] with parts {i, pi(i) + k}, where 1 <= i <= k. We also prove a similar jump, from exponential to factorial, in the possible speeds of monotone properties of ordered graphs, and of hereditary properties of ordered graphs not containing large complete, or complete bipartite ordered graphs. Our results generalize the Stanley-Wilf Conjecture on the number of n-permutations avoiding a fixed permutati…
This paper is a contribution to the study of hereditary classes of relational structures, these classes being quasi-ordered by embeddability. It deals with the specific case of ordered sets of width two and the corresponding bichains and incomparability graphs. Several open problems about hereditary classes of relational structures which have been considered over the years have positive answer in this case. For example, well-quasi-ordered hereditary classes of finite bipartite permutation graphs, respectively finite 321-avoiding permutations, have been characterized by Korpelainen, Lozin and M…
Let $\mathfrak{C}$ be a class of finite combinatorial structures. The \textit{profile} of $\mathfrak{C}$ is the function $φ_{\mathfrak{C}}$ which counts, for every integer $n$, the number $φ_{\mathfrak{C}}(n)$ of members of $\mathfrak{C}$ defined on $n$ elements, isomorphic structures been identified. The \textit{generating function of} $\mathfrak{C}$ is $\mathcal {H}_{\mathfrak{C}}(x):=\sum_{n\geqq 0}φ_{\mathfrak{C}}(n)x^{n}$. Many results about the behavior of the function $φ_{\mathfrak{C}}$ have been obtained. Albert and Atkinson have shown that the generating series of several classes of p…
Balogh, Bollobás and Morris (2006) have described a threshold phenomenon in the behavior of the profile of hereditary classes of ordered graphs. In this paper, we give an other look at their result based on the notion of monomorphic decomposition of a relational structure introduced in \cite{P-T-2013}. We prove that the class $\mathfrak S$ of ordered binary structures which do not have a finite monomorphic decomposition has a finite basis (a subset $\mathfrak A$ such that every member of $\mathfrak S$ embeds some member of $\mathfrak A$). In the case of ordered reflexive directed graphs, the b…
We collect some general results on graph limits associated to hereditary classes of graphs. As examples, we consider some classes defined by forbidden subgraphs and some classes of intersection graphs, including triangle-free graphs, chordal graphs, cographs, interval graphs, unit interval graphs, threshold graphs, and line graphs.
This note answers extremal questions like: what is the maximum number of edges in a graph of order n, which belongs to some hereditary property. The same question is answered also for the spectral radius and other similar parameters.
Each hereditary property can be characterized by its set of minimal obstructions; these sets are often unknown, or known but infinite. By allowing extra structure it is sometimes possible to describe such properties by a finite set of forbidden objects. This has been studied most intensely when the extra structure is a linear ordering of the vertex set. For instance, it is known that a graph G is $k$-colourable if and only if $V(G)$ admits a linear ordering $\le$ with no vertices $v_1 \le \cdots \le v_{k+1}$ such that $v_i v_{i+1} \in E(G)$ for every $i \in \{ 1, \dots, k \}$. In this paper, w…
A family of graphs $\mathcal{F}$ is hereditary if $\mathcal{F}$ is closed under isomorphism and taking induced subgraphs. The speed of $\mathcal{F}$ is the sequence $\{|\mathcal{F}^n|\}_{n \in \mathbb{N}}$, where $\mathcal{F}^n$ denotes the set of graphs in $\mathcal{F}$ with the vertex set $[n]$. Alon, Balogh, Bollobás and Morris [The structure of almost all graphs in a hereditary property, JCTB 2011] gave a rough description of typical graphs in a hereditary family and used it to show for every proper hereditary family $\mathcal{F}$ there exist $\varepsilon>0$ and an integer $l \geq 1$ such …
Korpelainen, Lozin, and Razgon conjectured that a hereditary property of graphs which is well-quasi-ordered by the induced subgraph order and defined by only finitely many minimal forbidden induced subgraphs is labelled well-quasi-ordered, a notion stronger than that of $n$-well-quasi-order introduced by Pouzet in the 1970s. We present a counterexample to this conjecture. In fact, we exhibit a hereditary property of graphs which is well-quasi-ordered by the induced subgraph order and defined by finitely many minimal forbidden induced subgraphs yet is not $2$-well-quasi-ordered. This counterexa…
A hereditary property of graphs is a collection of graphs which is closed under taking induced subgraphs. The speed of ¶is the function n \mapsto |¶_n|, where ¶_n denotes the graphs of order n in ¶. It was shown by Alekseev, and by Bollobas and Thomason, that if ¶is a hereditary property of graphs then |¶_n| = 2^{(1 - 1/r + o(1))n^2/2}, where r = r(¶) \in \N is the so-called `colouring number' of ¶. However, their results tell us very little about the structure of a typical graph G \in ¶. In this paper we describe the structure of almost every graph in a hereditary property of graphs, ¶. As a …
We consider hereditary classes of graphs equipped with a total order. We provide multiple equivalent characterisations of those classes which have bounded twin-width. In particular, we prove a grid theorem for classes of ordered graphs which have unbounded twin-width. From this we derive that the model-checking problem for first-order logic is fixed-parameter tractable over a hereditary class of ordered graphs if, and -- under common complexity-theoretic assumptions -- only if the class has bounded twin-width. For hereditary classes of ordered graphs, we show that bounded twin-width is equival…
We consider properties of edge-colored vertex-ordered graphs, i.e., graphs with a totally ordered vertex set and a finite set of possible edge colors. We show that any hereditary property of such graphs is strongly testable, i.e., testable with a constant number of queries. We also explain how the proof can be adapted to show that any hereditary property of $2$-dimensional matrices over a finite alphabet (where row and column order is not ignored) is strongly testable. The first result generalizes the result of Alon and Shapira [FOCS'05, SICOMP'08], who showed that any hereditary graph propert…
We prove that for every non-trivial hereditary family of graphs ${\cal P}$ and for every fixed $p \in (0,1)$, the maximum possible number of edges in a subgraph of the random graph $G(n,p)$ which belongs to ${\cal P}$ is, with high probability, $$ \left(1-\frac{1}{k-1}+o(1)\right)p{n \choose 2}, $$ where $k$ is the minimum chromatic number of a graph that does not belong to ${\cal P}$.
We give characterizations of the structure and degree sequences of hereditary unigraphs, those graphs for which every induced subgraph is the unique realization of its degree sequence. The class of hereditary unigraphs properly contains the threshold and matrogenic graphs, and the characterizations presented here naturally generalize those known for these other classes of graphs. The degree sequence characterization of hereditary unigraphs makes use of the list of values $k$ for which the $k$th Erdős--Gallai inequality holds with equality for a graphic sequence. Using the canonical decompositi…
In the companion paper [Linear rank-width of distance-hereditary graphs I. A polynomial-time algorithm, Algorithmica 78(1):342--377, 2017], we presented a characterization of the linear rank-width of distance-hereditary graphs, from which we derived an algorithm to compute it in polynomial time. In this paper, we investigate structural properties of distance-hereditary graphs based on this characterization. First, we prove that for a fixed tree $T$, every distance-hereditary graph of sufficiently large linear rank-width contains a vertex-minor isomorphic to $T$. We extend this property to bigg…
Theory of relations is the framework of this thesis. It is about enumeration of finite structures. Let $\mathscr C$ be a class of finite combinatorial structures, the \emph{profile} of $\mathscr C$ is the function $φ_{\mathscr C}$ which count, for every $n$, the number of members of $\mathscr{C}$ defined on $n$ elements, isomorphic structures been identified. The generating function for $\mathscr C$ is $\mathcal H_{\mathscr C}(X):=\sum_{n\geqq 0}φ_{\mathscr C}(n)X^n$. Many results about the behavior of the function $φ_{\mathscr C}$ have been obtained. Albert and Atkinson have shown that the ge…
Let $G$ be a graph on $n\geq 3$ vertices. A graph $G$ is almost distance-hereditary if each connected induced subgraph $H$ of $G$ has the property $d_{H}(x,y)\leq d_{G}(x,y)+1$ for any pair of vertices $x,y\in V(H)$. A graph $G$ is called 1-heavy (2-heavy) if at least one (two) of the end vertices of each induced subgraph of $G$ isomorphic to $K_{1,3}$ (a claw) has (have) degree at least $n/2$, and called claw-heavy if each claw of $G$ has a pair of end vertices with degree sum at least $n$. Thus every 2-heavy graph is claw-heavy. In this paper we prove the following two results: (1) Every 2-c…
An additive hereditary graph property is a set of graphs, closed under isomorphism and under taking subgraphs and disjoint unions. Let ${\cal P}_1, >..., {\cal P}_n$ be additive hereditary graph properties. A graph $G$ has property $({\cal P}_1 \circ ... \circ {\cal P}_n)$ if there is a partition $(V_1, ..., V_n)$ of $V(G)$ into $n$ sets such that, for all $i$, the induced subgraph $G[V_i]$ is in ${\cal P}_i$. A property ${\cal P}$ is reducible if there are properties ${\cal Q}$, ${\cal R}$ such that ${\cal P} = {\cal Q} \circ {\cal R}$; otherwise it is irreducible. Mihók, Semanišin and Vasky …
The paper [J. Balogh, B. Bollobás, D. Weinreich, A jump to the Bell number for hereditary graph properties, J. Combin. Theory Ser. B 95 (2005) 29--48] identifies a jump in the speed of hereditary graph properties to the Bell number $B_n$ and provides a partial characterisation of the family of minimal classes whose speed is at least $B_n$. In the present paper, we give a complete characterisation of this family. Since this family is infinite, the decidability of the problem of determining if the speed of a hereditary property is above or below the Bell number is questionable. We answer this qu…
A hereditary property of combinatorial structures is a collection of structures (e.g. graphs, posets) which is closed under isomorphism, closed under taking induced substructures (e.g. induced subgraphs), and contains arbitrarily large structures. Given a property P, we write P_n for the collection of distinct (i.e., non-isomorphic) structures in a property P with n vertices, and call the function n -> |P_n| the speed (or unlabelled speed) of P. Also, we write P^n for the collection of distinct labelled structures in P with vertices labelled 1,...,n, and call the function n -> |P^n| the labell…
We establish a list of characterizations of bounded twin-width for hereditary, totally ordered binary structures. This has several consequences. First, it allows us to show that a (hereditary) class of matrices over a finite alphabet either contains at least $n!$ matrices of size $n \times n$, or at most $c^n$ for some constant $c$. This generalizes the celebrated Stanley-Wilf conjecture/Marcus-Tardos theorem from permutation classes to any matrix class over a finite alphabet, answers our small conjecture [SODA '21] in the case of ordered graphs, and with more work, settles a question first as…
A graph property (i.e., a set of graphs) is induced-hereditary or additive if it is closed under taking induced-subgraphs or disjoint unions. If $\cP$ and $\cQ$ are properties, the product $\cP \circ \cQ$ consists of all graphs $G$ for which there is a partition of the vertex set of $G$ into (possibly empty) subsets $A$ and $B$ with $G[A] \in \cP$ and $G[B] \in \cQ$. A property is reducible if it is the product of two other properties, and irreducible otherwise. We completely describe the few reducible induced-hereditary properties that have a unique factorisation into irreducibles. Analogs of…
The vertex-nullity interlace polynomial of a graph, described by Arratia, Bollobás and Sorkin as evolving from questions of DNA sequencing, and extended to a two-variable interlace polynomial by the same authors, evokes many open questions. These include relations between the interlace polynomial and the Tutte polynomial and the computational complexity of the vertex-nullity interlace polynomial. Here, we prove that the one-variable vertex-nullity interlace polynomial is in general #P-hard to compute. We also show a relation between the two-variable interlace polynomial and the topological Tut…
Can the vertices of a graph $G$ be partitioned into $A \cup B$, so that $G[A]$ is a line-graph and $G[B]$ is a forest? Can $G$ be partitioned into a planar graph and a perfect graph? The NP-completeness of these problems are just special cases of our result: if ${\cal P}$ and ${\cal Q}$ are additive induced-hereditary graph properties, then $({\cal P}, {\cal Q})$-colouring is NP-hard, with the sole exception of graph 2-colouring (the case where both $\cal P$ and $\cal Q$ are the set ${\cal O}$ of finite edgeless graphs). Moreover, $({\cal P}, {\cal Q})$-colouring is NP-complete iff ${\cal P}$-…
As an application of Szemeredi's regularity lemma, Erdos-Frankl-Rodl (1986) showed that the number of graphs on vertex set {1,2,...n} with a monotone class P is $2^{(1+o(1))ex(n,P)n^2/2}$ where $ex(n,P)$ is the maximum number of edges of an n-vertex graph which has no subgraph in P. Kohayakawa et al. (2003) extended it from monotone to hereditary and from graphs to 3-uniform hypergraphs. We extend it to general hypergraphs. This may be a simple example illustrating how to apply a recent hypergraph regularity lemma by the author.
Let $H$ be a graph and $t\geq s\geq 2$ be integers. We prove that if $G$ is an $n$-vertex graph with no copy of $H$ and no induced copy of $K_{s,t}$, then $λ(G) = O\left(n^{1-1/s}\right)$ where $λ(G)$ is the spectral radius of the adjacency matrix of $G$. Our results are motivated by results of Babai, Guiduli, and Nikiforov bounding the maximum spectral radius of a graph with no copy (not necessarily induced) of $K_{s,t}$.
We give a complete characterization of bipartite graphs having tree-like Galois lattices. We prove that the poset obtained by deleting bottom and top elements from the Galois lattice of a bipartite graph is tree-like if and only if the graph is a Bipartite Distance Hereditary graph. By relying on the interplay between bipartite distance hereditary graphs and series-parallel graphs, we show that the lattice can be realized as the containment relation among directed paths in an arborescence. Moreover, a compact encoding of Bipartite Distance Hereditary graphs is proposed, that allows optimal tim…
Recent works of Alon-Shapira and Rödl-Schacht have demonstrated that every hereditary property of undirected graphs or hypergraphs is testable with one-sided error; informally, this means that if a graph or hypergraph satisfies that property "locally" with sufficiently high probability, then it can be perturbed (or "repaired") into a graph or hypergraph which satisfies that property "globally". In this paper we make some refinements to these results, some of which may be surprising. In the positive direction, we strengthen the results to cover hereditary properties of multiple directed polychr…
We characterize a large subclass of the class of those graphs $G$ for which the exponential domination number of $H$ equals the domination number of $H$ for every induced subgraph $H$ of $G$.
If P is a hereditary property then we show that, for the existence of a perfect f-factor, P is a sufficient condition for countable graphs and yields a sufficient condition for graphs of size aleph_1. Further we give two examples of a hereditary property which is even necessary for the existence of a perfect f-factor. We also discuss the aleph_2-case.
A graph $G$ is $H$-free if it does not contain an induced subgraph isomorphic to $H$. The study of the typical structure of $H$-free graphs was initiated by Erdős, Kleitman and Rothschild, who have shown that almost all $C_3$-free graphs are bipartite. Since then the typical structure of $H$-free graphs has been determined for several families of graphs $H$, including complete graphs, trees and cycles. Recently, Reed and Scott proposed a conjectural description of the typical structure of $H$-free graphs for all graphs $H$, which extends all previously known results in the area. We construct a…
The clique-width is known to be unbounded in the class of unit interval graphs. In this paper, we show that this is a minimal hereditary class of unbounded clique-width, i.e., in every hereditary subclass of unit interval graphs the clique-width is bounded by a constant.
Given a graph G and integers b and w. The black-and-white coloring problem asks if there exist disjoint sets of vertices B and W with |B|=b and |W|=w such that no vertex in B is adjacent to any vertex in W. In this paper we show that the problem is polynomial when restricted to cographs, distance-hereditary graphs, interval graphs and strongly chordal graphs. We show that the problem is NP-complete on splitgraphs.
A quick proof of Gallai's celebrated theorem on color-critical graphs is given from Gallai's simple, ingenious lemma on factor-critical graphs, in terms of partitioning the vertex-set into a minimum number of hyperedges of a hereditary hypergraph, generalizing the chromatic number. We then show examples of applying the results to new problems and indicate the way to algorithms and refined complexity results for all these examples at the same time.
A cycle-transversal of a graph G is a subset T of V(G) such that T intersects every cycle of G. A clique cycle-transversal, or cct for short, is a cycle-transversal which is a clique. Recognizing graphs which admit a cct can be done in polynomial time; however, no structural characterization of such graphs is known. We characterize distance-hereditary graphs admitting a cct in terms of forbidden induced subgraphs. This extends similar results for chordal graphs and cographs.
A graph is said to be distance-hereditary if the distance function in every connected induced subgraph is the same as in the graph itself. We prove that the ordinary Weisfeiler-Leman algorithm correctly tests the isomorphism of any two graphs if one of them is distance-hereditary; more precisely, the Weisfeiler-Leman dimension of the class of finite distance-hereditary graphs is equal to $2$. The previously best known upper bound for the dimension was $7$.
In this paper we consider certain types of betweenness axioms on the interval function $I_G$ of a connected graph $G$. We characterize the class of graphs for which $I_G$ satisfy these axioms. The class of graphs that we characterize include the important class of Ptolemaic graphs and some proper superclasses of Ptolemaic graphs: the distance hereditary graphs and the bridged graphs. We also provide axiomatic characterizations of the interval function of these classes of graphs using an arbitrary function known as \emph{transit function}.
Balogh, Bollobas and Weinreich showed that a parameter that has since been termed the distinguishing number can be used to identify a jump in the possible speeds of hereditary classes of graphs at the sequence of Bell numbers. We prove that every hereditary class that lies above the Bell numbers and has finite distinguishing number contains a boundary class for well-quasi-ordering. This means that any such hereditary class which in addition is defined by finitely many minimal forbidden induced subgraphs must contain an infinite antichain. As all hereditary classes below the Bell numbers are we…
Distance-hereditary graphs form an important class of graphs, from the theoretical point of view, due to the fact that they are the totally decomposable graphs for the split-decomposition. The previous best enumerative result for these graphs is from Nakano et al. (J. Comp. Sci. Tech., 2007), who have proven that the number of distance-hereditary graphs on $n$ vertices is bounded by ${2^{\lceil 3.59n\rceil}}$. In this paper, using classical tools of enumerative combinatorics, we improve on this result by providing an exact enumeration of distance-hereditary graphs, which allows to show that th…
Let $\mathcal{F}$ be a collection of $r$-uniform hypergraphs, and let $0 < p < 1$. It is known that there exists $c = c(p,\mathcal{F})$ such that the probability of a random $r$-graph in $G(n,p)$ not containing an induced subgraph from $\mathcal{F}$ is $2^{(-c+o(1)){n \choose r}}$. Let each graph in $\mathcal{F}$ have at least $t$ vertices. We show that in fact for every $ε> 0$, there exists $δ= δ(ε, p,\mathcal{F}) > 0$ such that the probability of a random $r$-graph in $G(n,p)$ containing less than $δn^t$ induced subgraphs each lying in $\mathcal{F}$ is at most $2^{(-c+ε){n \choose r}}$. This…
This paper is a contribution to the study of hereditary classes of finite graphs. We classify these classes according to the number of prime structures they contain. We consider such classes that are \emph{minimal prime}: classes that contain infinitely many primes but every proper hereditary subclass contains only finitely many primes. We give a complete characterization of such classes. In fact, each one of these classes is a well quasi ordered age and there are uncountably many of them. Eleven of these ages remain well quasi ordered when labels in a well quasi ordering are added. Among the …
Bandelt and Mulder's structural characterization of Bipartite Distance Hereditary graphs asserts that such graphs can be built inductively starting from a single vertex and by repeatedly adding either pending vertices or twins (i.e., vertices with the same neighborhood as an existing one). Dirac and Duffin's structural characterization of 2-connected series-parallel graphs asserts that such graphs can be built inductively starting from a single edge by adding either edges in series or in parallel. In this paper we prove that the two constructions are the same construction when bipartite graphs…
Given a family of graphs $\mathcal{F}$, we prove that the normalized edit distance of any given graph $Γ$ to being induced $\mathcal{F}$-free is estimable with a query complexity that depends only on the bounds of the Frieze--Kannan Regularity Lemma and on a Removal Lemma for $\mathcal{F}$.
The edge-connectivity of a graph is the minimum number of edges whose deletion disconnects the graph. Let $Δ(G)$ the maximum degree of a graph $G$ and let $ρ(G)$ be the spectral radius of $G$. In this article we present a lower bound for $Δ(G)-ρ(G)$ in terms of the edge connectivity of $G$, where $G$ is a nonregular distance-hereditary graph. We also prove that $ρ(G)$ reaches the maximum at a unique graph in $\mathcal G$, when $\vert V(G)\vert = n$, and $\mathcal G$ either is in the class of graphs with bounded tree-width or is in the class of block graphs with prescribed independence number.
We study the computational complexity of computing role colourings of graphs in hereditary classes. We are interested in describing the family of hereditary classes on which a role colouring with k colours can be computed in polynomial time. In particular, we wish to describe the boundary between the "hard" and "easy" classes. The notion of a boundary class has been introduced by Alekseev in order to study such boundaries. Our main results are a boundary class for the k-role colouring problem and the related k-coupon colouring problem which has recently received a lot of attention in the liter…
A König-Egerváry graph is a graph $G$ satisfying $α(G)+μ(G)=|V(G)|$, where $α(G)$ is the cardinality of a maximum independent set and $μ(G)$ is the matching number of $G$. Such graphs are those that admit a matching between $V(G)-\bigcup Γ$ and $\bigcap Γ$ where $Γ$ is a set-system comprised of maximum independent sets satisfying $|\bigcup Γ'|+|\bigcap Γ'|=2α(G)$ for every set-system $Γ' \subseteq Γ$; in order to improve this characterization of a König-Egerváry graph, we characterize \emph{hereditary König-Egerváry set-systems} (HKE set-systems, here after). An \emph{HKE} set-system is a set-…
