Svante Janson
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.
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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.
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 …
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 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…
Representing distance-hereditary graphs with multi-rooted trees
本文提出树状网络这一根树的推广概念，证明距离遗传图类可由带标签树状网络解释，类似余图类由带标签根树解释，建立了距离遗传图与多根树结构的对应关系。
A note on distance-hereditary graphs whose complement is also distance-hereditary
本文通过分裂分解和模分解研究距离-hereditary图的补图性质，揭示其结构特征和分类方法。
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…
The number of trees in distance-hereditary graphs and their friends
本文统一证明并推广了2020年代关于距离遗传图生成树计数的若干结果，通过研究顶点复制操作下度枚举子的行为，并证明了Ehrenborg猜想与其多项式形式的等价性。
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 in…
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 …
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…
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…
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…
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…
Hereditary Nordhaus-Gaddum Graphs
该研究对Nordhaus-Gaddum图进行遗传性推广，探究所有诱导子图均满足色数和不等式的图类，刻画其禁用诱导子图、与常见图类的交集，并讨论χ-有界性与算法结果。
We prove an arithmetic analogue of the typical structure theorem for graph hereditary properties due to Alon, Balogh, Bollobás and Morris.
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 ${\cal P}$-…
The Hereditary Closure of the Unigraphs
本文针对单图类不具遗传性的问题，研究其遗传闭包HCU类，借助Tyshkevich分解与Rao度序列偏序从多方面刻画HCU，并给出诱导子图均为单图的单图类的新刻画。
Subgraphs of random graphs in hereditary families
研究遗传族中随机图的子图最大边数，证明对非平凡遗传性质，随机图G(n,p)中属于该性质的子图边数以高概率不超过n^{2-ε}，回答了Alon等人的问题。
On the distinguishing chromatic number in hereditary graph classes
本文研究遗传图类中的区分色数，证明在禁止某些小图作为诱导子图时，区分色数的上界可显著降低。
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.
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…
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 $K_{s,t}$.
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.
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 $G$.
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 $7$.
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.
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}.
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…
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…
The undecidability of joint embedding and joint homomorphism for hereditary graph classes
本文通过平铺问题归约证明了遗传图类的联合嵌入性质及联合同态性质均为不可判定，为图类理论提供了新的不可判定性结果。
On the analogue of Esperet's conjecture: Characterizing hereditary classes
本文研究了埃斯佩特猜想的类比，探讨了继承图类中是否存在多项式χ-有界函数的条件。
The class of cographs is known to have unbounded linear clique-width. We prove that a hereditary class of cographs has bounded linear clique-width if and only if it does not contain all quasi-threshold graphs or their complements. The proof borrows ideas from the enumeration of permutation classes.
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.
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…
Edge-apexing in hereditary classes of graphs
本文证明遗传图类在边顶点化操作下保持有限禁止诱导子图性质，并针对余图类给出禁止诱导子图阶数上界为8且通过计算机搜索全部枚举。
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.
A graph $G=(V,E)$ is distance hereditary if every induced path of $G$ is a shortest path. In this paper, we show that the eccentricity function $e(v)=\max\{d(v,u): u\in V\}$ in any distance-hereditary graph $G$ is almost unimodal, that is, every vertex $v$ with $e(v)> rad(G)+1$ has a neighbor with smaller eccentricity. Here, $rad(G)=\min\{e(v): v\in V\}$ is the radius of graph $G$. Moreover, we use this result to fully characterize the centers of distance-hereditary graphs. Several bounds on the eccentricity of a vertex with respect to its distance to the center of $G$ or to the ends of a diam…
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…
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…
Characterization and linear-time recognition of balanced distance-hereditary graphs
本文证明在距离遗传图中，平衡图等价于遗传团-Helly图，并由单个禁止诱导子图$\overline{3K_2}$刻画，进而给出线性时间识别算法。
The asymptotic $χ$-boundedness of hereditary families
本文探讨了遗传图族渐近 $χ$-有界的问题，证明了对于所有树和除 $C_6$ 外的环，其无导出子图族几乎都满足 $χ(G)=ω(G)$，并给出了 $C_6$-free 图的渐近 $χ$-有界函数。
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-…
Hereditary pattern-free classes are not always 2-wqo
研究遗传无模式类与2-良拟序的关系，核心方法是展示特定无模式遗传图类，主要贡献是回答相关论文问题，表明存在非2-良拟序的此类图类。
The speed of a hereditary property $P$ is the number $P_n$ of $n$-vertex labelled graphs in $P$. It is known that the rates of growth of $P_n$ constitute discrete layers and the speed jumps, in particular, from constant to polynomial, from polynomial to exponential and from exponential to factorial. One more jump occurs when the entropy $\lim_{n\to\infty}\frac{\log_2 P_n}{\binom{n}{2}}$ changes from 0 to a nonzero value. In the present paper, for each of these jumps we identify a graph parameter responsible for it, i.e. we show that a jump of the speed coincides with a jump of the respective p…
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.
Limits of graphs were initiated recently in the two extreme contexts of dense and bounded degree graphs. This led to elegant limiting structures called graphons and graphings. These approach have been unified and generalized by authors in a more general setting using a combination of analytic tools and model theory to FO-limits (and X-limits) and to the notion of modeling. The existence of modeling limits was established for sequences in a bounded degree class and, in addition, to the case of classes of trees with bounded height and of graphs with bounded tree depth. These seemingly very speci…
Given a finite relational language $\calL$, a hereditary $\calL$-property is a class of finite $\calL$-structures which is closed under isomorphism and model theoretic substructure. This notion encompasses many objects of study in extremal combinatorics, including (but not limited to) hereditary properties of graphs, hypergraphs, and oriented graphs. In this paper, we generalize certain definitions, tools, and results form the study of hereditary properties in combinatorics to the setting of hereditary $\calL$-properties, where $\calL$ is any finite relational language with maximum arity at le…
A non-hereditary Pollyanna class that is not strongly Pollyanna
本文构造了一个非遗传的Pollyanna类，该类不是强Pollyanna类，通过证明该类与任何k-good类的交集在χ-有界性上不满足多项式有界性。
Uncountably many minimal hereditary classes of graphs of unbounded clique-width
本研究通过字母表{0,1,2,3}上的无限字定义二部遗传图类，刻画其团宽无界及极小性的条件，构造出不可数个无界团宽的极小遗传图类，同时证明相关猜想的一个方向成立、另一方向不成立。
Circle graphs are intersection graphs of chords in a circle and $k$-polygon graphs are intersection graphs of chords in a convex $k$-sided polygon where each chord has its endpoints on distinct sides. The $k$-polygon graphs, for $k \ge 2$, form an infinite chain of graph classes, each of which contains the class of permutation graphs. The union of all of those graph classes is the class of circle graphs. The polygon number $\gp(G)$ of a circle graph $G$ is the minimum $k$ such that $G$ is a $k$-polygon graph. Given a circle graph $G$ and an integer $k$, determining whether $\gp(G) \le k$ is NP…
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