Hereditary Graph Property Research Articles

Fractional Q-coloring of graphs

An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let Q be an additive hereditary property of graphs. A Q-edge-coloring of a simple graph is an edge coloring in which the edges colored with the same color induce a subgraph of property Q. In this paper we present some results on fractional Q-edge-colorings. We determine the fractional Q-edge chromatic number for matroidal properties of graphs.

The property of [formula omitted]-colourable graphs is uniquely decomposable

An additive hereditary graph property is a class of simple graphs which is closed under unions, subgraphs and isomorphisms. If P1,…,Pn are graph properties, then a (P1,…,Pn)-decomposition of a graph G is a partition E1,…,En of E(G) such that G[Ei], the subgraph of G induced by Ei, is in Pi, for i=1,…,n. The sum of the properties P1,…,Pn is the property P1⊕⋯⊕Pn={G∈I:G has a (P1,…,Pn)-decomposition}. A property P is said to be decomposable if there exist non-trivial additive hereditary properties P1 and P2 such that P=P1⊕P2. A property is uniquely decomposable if, apart from the order of the factors, it can be written as a sum of indecomposable properties in only one way. We show that not all properties are uniquely decomposable; however, the property of k-colourable graphs Ok is a uniquely decomposable property.

Formula omitted]-Ramsey classes of graphs

In this paper, we investigate the problem of (k,Q)-Ramsey classes of graphs, which were introduced in [M. Borowiecki, A. Fiedorowicz, On partitions of hereditary properties of graphs, Discuss. Math. Graph Theory 26 (2006) 377–387] as an extension of the well-known notion of a vertex Ramsey class of graphs. The definition is based on a concept of a (k;Q)-colouring of graphs. Let Q be a hereditary graph property and assume Q⊆O2, where O2 denotes the class of all bipartite graphs. A (k;Q)-colouring of a graph G is a mapping f from the set of vertices of G to the set {1,…,k} of colours, such that for any two distinct colours i and j, the subgraph induced in G by all the edges xy such that f(x)=i and f(y)=j has the property Q. A graph property P is called a (k,Q)-Ramsey class of graphs, if for any graph G∈P, there is a graph H∈P such that in any (k;Q)-colouring of H we have a monochromatic copy of G.We prove that some important graph classes, such as k-degenerate graphs, k-trees or hom-properties, are (k,Q)-Ramsey classes of graphs. We also provide sufficient conditions for a graph property to be a (k,Q)-Ramsey class.

Every hereditary permutation property is testable

Abstract In 2010 Moreira and three of the current authors introduced a notion of property testing for permutations based on a distance that measures the randomness of a permutation. They proved that every hereditary permutation property is testable in this sense, or weakly testable in their terminology. In this paper, we show that hereditary permutation properties are testable in a stronger sense, namely when testability is defined by means of the usual l 1 -distance for permutations. This is the permutation analogue of the well-known Alon-Shapira result on the testability of hereditary graph properties.

Unique factorization of compositive hereditary graph properties

A graph property is any class of graphs that is closed under isomorphisms. A graph property P is hereditary if it is closed under taking subgraphs; it is compositive if for any graphs G 1,G 2 ∈ P there exists a graph G ∈ P containing both G 1 and G 2 as subgraphs.Let H be any given graph on vertices υ 1, ..., υ n , n ≥ 2. A graph property P is H-factorizable over the class of graph properties ℙ if there exist P 1, ..., P n ∈ ℙ such that P consists of all graphs whose vertex sets can be partitioned into n parts, possibly empty, satisfying: 1. for each i, the graph induced by the i-th non-empty partition part is in P i , and 2. for each i and j with i ≠ j, there is no edge between the i-th and j-th parts if υ i and υ j are non-adjacent vertices in H. If a graph property P is H-factorizable over ℙ and we know the graph properties P 1, ..., P n , then we write P = H[P 1, ..., P n ]. In such a case, the presentation H[P 1, ..., P n ] is called a factorization of P over ℙ. This concept generalizes graph homomorphisms and (P 1, ..., P n )-colorings.In this paper, we investigate all H-factorizations of a graph property P over the class of all hereditary compositive graph properties for finite graphs H. It is shown that in many cases there is exactly one such factorization.

A Note on the Speed of Hereditary Graph Properties

For a graph property $X$, let $X_n$ be the number of graphs with vertex set $\{1,\ldots,n\}$ having property $X$, also known as the speed of $X$. A property $X$ is called factorial if $X$ is hereditary (i.e. closed under taking induced subgraphs) and $n^{c_1n}\le X_n\le n^{c_2n}$ for some positive constants $c_1$ and $c_2$. Hereditary properties with the speed slower than factorial are surprisingly well structured. The situation with factorial properties is more complicated and less explored, although this family includes many properties of theoretical or practical importance, such as planar graphs or graphs of bounded vertex degree. To simplify the study of factorial properties, we propose the following conjecture: the speed of a hereditary property $X$ is factorial if and only if the fastest of the following three properties is factorial: bipartite graphs in $X$, co-bipartite graphs in $X$ and split graphs in $X$. In this note, we verify the conjecture for hereditary properties defined by forbidden induced subgraphs with at most 4 vertices.

Testing permutation properties through subpermutations

There has been great interest in deciding whether a combinatorial structure satisfies some property, or in estimating the value of some numerical function associated with this combinatorial structure, by considering only a randomly chosen substructure of sufficiently large, but constant size. These problems are called property testing and parameter testing, where a property or parameter is said to be testable if it can be estimated accurately in this way. The algorithmic appeal is evident, as, conditional on sampling, this leads to reliable constant-time randomized estimators. Our paper addresses property testing and parameter testing for permutations in a subpermutation perspective; more precisely, we investigate permutation properties and parameters that can be well approximated based on a randomly chosen subpermutation of much smaller size. In this context, we use a theory of convergence of permutation sequences developed by the present authors [C. Hoppen, Y. Kohayakawa, C.G. Moreira, R.M. Sampaio, Limits of permutation sequences through permutation regularity, Manuscript, 2010, 34pp.] to characterize testable permutation parameters along the lines of the work of Borgs et al. [C. Borgs, J. Chayes, L. Lovász, V.T. Sós, B. Szegedy, K. Vesztergombi, Graph limits and parameter testing, in: STOC’06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, ACM, New York, 2006, pp. 261–270.] in the case of graphs. Moreover, we obtain a permutation result in the direction of a famous result of Alon and Shapira [N. Alon, A. Shapira, A characterization of the (natural) graph properties testable with one-sided error, SIAM J. Comput. 37 (6) (2008) 1703–1727.] stating that every hereditary graph property is testable.

Generalized total colorings of graphs

An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphism. Let P and Q be additive hereditary properties of graphs. A (P ,Q)-total coloring ∗Research supported in part by Slovak VEGA Grant 2/0194/10. 210 M. Borowiecki, A. Kemnitz, M. Marangio and P. Mihok of a simple graphG is a coloring of the vertices V (G) and edgesE(G) of G such that for each color i the vertices colored by i induce a subgraph of property P , the edges colored by i induce a subgraph of property Q and incident vertices and edges obtain different colors. In this paper we present some general basic results on (P ,Q)-total colorings. We determine the (P ,Q)-total chromatic number of paths and cycles and, for specific properties, of complete graphs. Moreover, we prove a compactness theorem for (P ,Q)-total colorings.

The structure of almost all graphs in a hereditary property

A hereditary property of graphs is a collection of graphs which is closed under taking induced subgraphs. The speed of P is the function n ↦ | P n | , where P n denotes the graphs of order n in P . It was shown by Alekseev, and by Bollobás and Thomason, that if P is a hereditary property of graphs then | P n | = 2 ( 1 − 1 / r + o ( 1 ) ) ( n 2 ) , where r = r ( P ) ∈ N is the so-called ‘colouring number’ of P . However, their results tell us very little about the structure of a typical graph G ∈ P . In this paper we describe the structure of almost every graph in a hereditary property of graphs, P . As a consequence, we derive essentially optimal bounds on the speed of P , improving the Alekseev–Bollobás–Thomason Theorem, and also generalising results of Balogh, Bollobás and Simonovits.

Hardness of edge-modification problems

For a graph property P consider the following computational problem. Given an input graph G , what is the minimum number of edge modifications (additions and/or deletions) that one has to apply to G in order to turn it into a graph that satisfies P ? Namely, what is the edit distance Δ ( G , P ) of a graph G from satisfying P ? Clearly, the computational complexity of such a problem strongly depends on P . For over 30 years this family of computational problems has been studied in several contexts and various algorithms, as well as hardness results, were obtained for specific graph properties. Alon, Shapira and Sudakov studied in [N. Alon, A. Shapira, B. Sudakov, Additive approximation for edge-deletion problems, in: Proc. of the 46th IEEE FOCS, 2005, 419–428. Also: Annals of Mathematics (in press)] the approximability of the computational problem for the family of monotone graph properties, namely properties that are closed under removal of edges and vertices. They describe an efficient algorithm that achieves an o ( n 2 ) additive approximation to Δ ( G , P ) for any monotone property P , where G is an n -vertex input graph, and show that the problem of achieving an O ( n 2 − ε ) additive approximation is N P -hard for most monotone properties. The methods in [N. Alon, A. Shapira, B. Sudakov, Additive approximation for edge-deletion problems, in: Proc. of the 46th IEEE FOCS, 2005, 419–428. Also: Annals of Mathematics (in press)] also provide a polynomial time approximation algorithm which computes Δ ( G , P ) ± o ( n 2 ) for the broader family of hereditary graph properties (which are closed under removal of vertices). In this work we introduce two approaches for showing that improving upon the additive approximation achieved by this algorithm is N P -hard for several sub-families of hereditary properties. In addition, we state a conjecture on the hardness of computing the edit distance from being induced H -free for any forbidden graph H .

Testing Hereditary Properties of Nonexpanding Bounded-Degree Graphs

We study graph properties that are testable for bounded-degree graphs in time independent of the input size. Our goal is to distinguish between graphs having a predetermined graph property and graphs that are far from every graph having that property. It is well known that in the bounded-degree graph model (where two graphs are considered “far” if they differ in $\varepsilon n$ edges for a positive constant $\varepsilon$), many graph properties cannot be tested even with a constant or even with a polylogarithmic number of queries. Therefore in this paper we focus our attention on testing graph properties for special classes of graphs. Specifically, we show that every hereditary graph property is testable with a constant number of queries provided that every sufficiently large induced subgraph of the input graph has poor expansion. This result implies that, for example, any hereditary property (e.g., $k$-colorability, $H$-freeness, etc.) is testable in the bounded-degree graph model for planar graphs, graphs with bounded genus, interval graphs, etc. No such results have been known before, and prior to our work, very few graph properties have been known to be testable with a constant number of queries for general graph classes in the bounded-degree graph model.

On infinite uniquely partitionable graphs and graph properties of finite character

A graph property is any nonempty isomorphism-closed class of simple (finite or infinite) graphs. A graph property P is of finite character if a graph G has a property P if and only if every finite induced subgraph of G has a property P . Let P1,P2, . . . ,Pn be graph properties of finite character, a graph G is said to be (uniquely) (P1,P2, . . . ,Pn)partitionable if there is an (exactly one) partition {V1, V2, . . . , Vn} of V (G) such that G[Vi] ∈ Pi for i = 1, 2, . . . , n. Let us denote by R = P1◦P2◦ · · · ◦Pn the class of all (P1,P2, . . . ,Pn)-partitionable graphs. A property R = P1◦P2◦ · · · ◦Pn, n ≥ 2 is said to be reducible. We prove that any reducible additive graph property R of finite character has a uniquely (P1,P2, . . . ,Pn)-partitionable countable generating graph. We also prove that for a reducible additive hereditary graph property R of finite character there exists a weakly universal countable graph if and only if each property Pi has a weakly universal graph. 242 J. Bucko and P. Mihok

Cardinality of a minimal forbidden graph family for reducible additive hereditary graph properties

An additive hereditary graph property is any class of simple graphs, which is closed under isomorphisms unions and taking subgraphs. Let L a denote a class of all such properties. In the paper, we consider Hreducible over L a properties with H being a xed graph. The niteness of the sets of all minimal forbidden graphs is analyzed for such properties.

The unlabelled speed of a hereditary graph property

A property of graphs is a collection P of graphs closed under isomorphism; we call P hereditary if it is closed under taking induced subgraphs. Given a property P , we write P n for the set of graphs in P with vertex set [ n ] = { 1 , … , n } , and P n for the isomorphism classes of graphs of order n that are in P . The cardinality | P n | is the labelled speed of P and | P n | is the unlabelled speed. In the last decade numerous results have been proved about the labelled speeds of hereditary properties, with emphasis on the striking phenomenon that only certain speeds are possible: there are various pairs of functions ( f ( n ) , F ( n ) ) , with F ( n ) much larger than f ( n ) , such that if the labelled speed is infinitely often larger than f ( n ) then it is also larger than F ( n ) for all sufficiently large values of n. Putting it concisely: the speed jumps from f ( n ) to F ( n ) . Recent work on hereditary graph properties has shown that “large” and “small” labelled speeds of hereditary graph properties do jump. The aim of this paper is to study the unlabelled speed of a hereditary property, with emphasis on jumps. Among other results, we shall show that the unlabelled speed of a hereditary graph property is either of polynomial order or at least S ( n ) , the number of ways of partitioning a set with n indistinguishable elements.

What is the furthest graph from a hereditary property?

AbstractFor a graph property P, the edit distance of a graph G from P, denoted EP(G), is the minimum number of edge modifications (additions or deletions) one needs to apply to G to turn it into a graph satisfying P. What is the furthest graph on n vertices from P and what is the largest possible edit distance from P? Denote this maximal distance by ed(n,P). This question is motivated by algorithmic edge‐modification problems, in which one wishes to find or approximate the value of EP(G) given an input graph G. A monotone graph property is closed under removal of edges and vertices. Trivially, for any monotone property, the largest edit distance is attained by a complete graph. We show that this is a simple instance of a much broader phenomenon. A hereditary graph property is closed under removal of vertices. We prove that for any hereditary graph property P, a random graph with an edge density that depends on P essentially achieves the maximal distance from P, that is: ed(n,P) = EP(G(n,p(P))) + o(n2) with high probability. The proofs combine several tools, including strengthened versions of the Szemerédi regularity lemma, properties of random graphs and probabilistic arguments. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008

A Characterization of the (Natural) Graph Properties Testable with One-Sided Error

The problem of characterizing all the testable graph properties is considered by many to be the most important open problem in the area of property testing. Our main result in this paper is a solution of an important special case of this general problem: Call a property tester oblivious if its decisions are independent of the size of the input graph. We show that a graph property ${\cal P}$ has an oblivious one-sided error tester if and only if ${\cal P}$ is semihereditary. We stress that any “natural” property that can be tested (either with one-sided or with two-sided error) can be tested by an oblivious tester. In particular, all the testers studied thus far in the literature were oblivious. Our main result can thus be considered as a precise characterization of the natural graph properties, which are testable with one-sided error. One of the main technical contributions of this paper is in showing that any hereditary graph property can be tested with one-sided error. This general result contains as a special case all the previous results about testing graph properties with one-sided error. More importantly, as a special case of our main result, we infer that some of the most well-studied graph properties, both in graph theory and computer science, are testable with one-sided error. Some of these properties are the well-known graph properties of being perfect, chordal, interval, comparability, permutation, and more. None of these properties was previously known to be testable.

The maximum edit distance from hereditary graph properties

For a graph property P , the edit distance of a graph G from P , denoted E P ( G ) , is the minimum number of edge modifications (additions or deletions) one needs to apply to G in order to turn it into a graph satisfying P . What is the largest possible edit distance of a graph on n vertices from P ? Denote this distance by ed ( n , P ) . A graph property is hereditary if it is closed under removal of vertices. In a previous work, the authors show that for any hereditary property, a random graph G ( n , p ( P ) ) essentially achieves the maximal distance from P , proving: ed ( n , P ) = E P ( G ( n , p ( P ) ) ) + o ( n 2 ) with high probability. The proof implicitly asserts the existence of such p ( P ) , but it does not supply a general tool for determining its value or the edit distance. In this paper, we determine the values of p ( P ) and ed ( n , P ) for some subfamilies of hereditary properties including sparse hereditary properties, complement invariant properties, ( r , s ) -colorability and more. We provide methods for analyzing the maximum edit distance from the graph properties of being induced H-free for some graphs H, and use it to show that in some natural cases G ( n , 1 / 2 ) is not the furthest graph. Throughout the paper, the various tools let us deduce the asymptotic maximum edit distance from some well studied hereditary graph properties, such as being Perfect, Chordal, Interval, Permutation, Claw-Free, Cograph and more. We also determine the edit distance of G ( n , 1 / 2 ) from any hereditary property, and investigate the behavior of E P ( G ( n , p ) ) as a function of p. The proofs combine several tools in Extremal Graph Theory, including strengthened versions of the Szemerédi Regularity Lemma, Ramsey Theory and properties of random graphs.

Hereditary properties of combinatorial structures: Posets and oriented graphs

AbstractA 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 $\cal {P}$, we write $\cal {P}_{n}$ for the collection of distinct (i.e., non‐isomorphic) structures in a property $\cal {P}$ with n vertices, and call the function ${n} \mapsto |\cal {P}_{n}|$ the speed (or unlabeled speed) of $\cal {P}$. Also, we write $\cal {P}^{n}$ for the collection of distinct labeled structures in $\cal {P}$ with vertices labeled $1,\ldots,{n}$, and call the function ${n} \mapsto |\cal {P}^{n}|$ the labeled speed of $\cal {P}$. The possible labeled speeds of a hereditary property of graphs have been extensively studied, and the aim of this article is to investigate the possible speeds of other combinatorial structures, namely posets and oriented graphs. More precisely, we show that (for sufficiently large n), the labeled speed of a hereditary property of posets is either 1, or exactly a polynomial, or at least $2^{n}-{1}$. We also show that there is an initial jump in the possible unlabeled speeds of hereditary properties of posets, tournaments, and directed graphs, from bounded to linear speed, and give a sharp lower bound on the possible linear speeds in each case. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 311–332, 2007

On invariants of hereditary graph properties

The product P ∘ Q of graph properties P , Q is the class of all graphs having a vertex-partition into two parts inducing subgraphs with properties P and Q , respectively. For a graph invariant ϕ and a graph property P we define ϕ ( P ) as the minimum of ϕ ( F ) taken over all minimal forbidden subgraphs F of P . An invariant of graph properties ϕ is said to be additive with respect to reducible hereditary properties if ϕ ( P ∘ Q ) = ϕ ( P ) + ϕ ( Q ) for every pair of hereditary properties P , Q . In this paper, we provide necessary and sufficient conditions for invariants to be additive with respect to reducible hereditary graph properties. We prove that the subchromatic number, the degeneracy number and tree-width and some other invariants of hereditary graph properties satisfy those conditions.

Generalized Matrogenic Graphs

Using a forbidden induced subsystem characterization of matroids, we propose a method for characterizing classes of \( \mathcal{P} \)-matrogenic graphs for all hereditary graph properties \( \mathcal{P} \).