Hereditary Classes Research Articles

On the König Graphs for a $$\boldsymbol 5 $$-Path and Its Spanning Supergraphs

We describe the hereditary class of graphs whose every subgraph has the property that the maximum number of disjoint $$5$$ -paths (paths on $$5 $$ vertices) is equal to the minimum size of the sets of vertices having nonempty intersection with the vertex set of each $$5 $$ -path. We describe this class in terms of the “forbidden subgraphs” and give an alternative description, using some operations on pseudographs.

$\mu$-Paracompactness via Hereditary Classes

The notion of paracompactness in generalized topology is introduced and studied in [5]. In this paper, we introduce and investigate the notion of $\mu$-paracompact spaces with respect to a hereditary class $\mathcal{H}$, which is a generalization of the notion of $\mu$-paracompact spaces. We study characterizations, subsets, and subspaces of $\mu\mathcal{H}$-paracompact spaces. Also, we investigate the invariants of $\mu\mathcal{H}$-paracompact spaces by functions.

Between clique-width and linear clique-width of bipartite graphs

We consider hereditary classes of bipartite graphs where clique-width is bounded, but linear clique-width is not. Our goal is identifying classes that are critical with respect to linear clique-width. We discover four such classes and conjecture that this list is complete, i.e. a hereditary class of bipartite graphs of bounded clique-width that excludes a graph from each of the four critical classes has bounded linear clique-width.

Factorization of maps on limits of inverse systems in Top

For some productive and closed hereditary classes C in Hausdorff topological spaces a cardinal κ will be obtained such that every continuous map on a limit of an inverse system in C into a given space can be factorized via a limit of a subsystem of cardinality less than κ. Large cardinals will be needed to get some general results.

A complete classification of the complexity of the vertex 3-colourability problem for quadruples of induced 5-vertex prohibitions

The vertex 3-colourability problem for a given graph is to check whether it is possible to split the set of its vertices into three subsets of pairwise non-adjacent vertices or not. A hereditary class of graphs is a set of simple graphs closed under isomorphism and deletion of vertices; the set of its forbidden induced subgraphs defines every such a class. For all but three the quadruples of 5-vertex forbidden induced subgraphs, we know the complexity status of the vertex 3-colourability problem. Additionally, two of these three cases are polynomially equivalent; they also polynomially reduce to the third one. In this paper, we prove that the computational complexity of the considered problem in all of the three mentioned classes is polynomial. This result contributes to the algorithmic graph theory.

2413-Balloon Permutations and the Growth of the Möbius Function

We show that the growth of the principal Möbius function on the permutation poset is at least exponential. This improves on previous work, which has shown that the growth is at least polynomial. We define a method of constructing a permutation from a smaller permutation which we call ``"ballooning". We show that if $\beta$ is a 2413-balloon, and $\pi$ is the 2413-balloon of $\beta$, then $\mu[1,\pi] = 2 \mu[1,\beta]$. This allows us to construct a sequence of permutations $\pi_1, \pi_2, \pi_3\ldots$ with lengths $n, n+4, n+8, \ldots$ such that $\mu[1,\pi_{i+1}] = 2 \mu[1,\pi_{i}]$, and this gives us exponential growth. Further, our construction method gives permutations that lie within a hereditary class with finitely many simple permutations. We also find an expression for the value of $\mu[1,\pi]$, where $\pi$ is a 2413-balloon, with no restriction on the permutation being ballooned.

Hereditary classes of ordered binary structures

Balogh, Bollobas 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 [32]. We prove that the class 𝔖 of ordered binary structures which do not have a finite monomorphic decomposition has a finite basis (a subset 𝔄 such that every member of 𝔖 embeds some member of 𝔄). In the case of ordered reflexive directed graphs, the basis has 1242 members and the profile of their ages grows at least as the Fibonacci function. From this result, we deduce that the following dichotomy property holds for every hereditary class ℭ of finite ordered binary structures of a given finite type. Either there is an integer l such that every member of ℭ has a monomorphic decomposition into at most l blocks and in this case the profile of ℭ is bounded by a polynomial of degree l − 1 (and in fact is a polynomial), or ℭ contains the age of a structure which does not have a finite monomorphic decomposition, in which case the profile of ℭ is bounded below by the Fibonacci function.

Big lattices of module classes induced by preradicals

We introduce new big lattices of classes of R-modules induced by suitable preradicals, the big lattice of $$\sigma $$-hereditary classes, the big lattice of$$\sigma $$-cohereditary classes, and the big lattice of $$\sigma $$-open classes. We prove that their Skeletons are boolean lattices which generalize and extend the lattice of natural classes, the lattice of conatural classes and the Skeleton of the lattice of hereditary torsion theories, respectively. We also introduce the lattice of$$\sigma $$-hereditary torsion theories, for a exact preradical $$\sigma $$. In case that $$\sigma $$ be the identity preradical, we obtain the usual lattice of hereditary torsion classes.

Well‐quasi‐ordering and finite distinguishing number

AbstractBalogh, Bollobás 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 well‐quasi‐ordered, our results complete the answer to the question of well‐quasi‐ordering for hereditary classes with finite distinguishing number. We also show that the decision procedure of Atminas, Collins, Foniok and Lozin to decide the Bell number (and which now also decides well‐quasi‐ordering for classes of finite distinguishing number) has runtime bounded by an explicit (quadruple exponential) function of the order of the largest minimal forbidden induced subgraph of the class.

Operators in minimal spaces with hereditary classes

Operators in minimal spaces with hereditary classes

Extension of GTS through hereditary class

In literature there are the notions of generalized topology (in short GT) and hereditary class. Both these notions have been studied extensively in different directions. Most of these studies evolved around considering a generalized topological space (in short GTS) endowed with certain hereditary class, and subsequently the nature of the resultant space is studied. By this, mathematicians intend to measure the quantum of deviations, the resultant space receives in respect of its basic properties. In our work we are deviating from this course of study. We consider an arbitrary GTS and certain collection of hereditary classes on this space and next we lift this collection to a GTS and the initial GTS will be embedded to this newly formulated space through certain well defined embedding map. Further we study the compactness property of this extension space. In course of this study we introduce certain notions and corresponding results for the theories of GTS as well as of hereditary class.

Composability of Permutation Classes

We define the operation of composing two hereditary classes of permutations using the standard composition of permutations as functions and we explore properties and structure of permutation classes considering this operation. We mostly concern ourselves with the problem of whether permutation classes can be composed from their proper subclasses. We provide examples of classes which can be composed from two proper subclasses, classes which can be composed from three but not from two proper subclasses and classes which cannot be composed from any finite number of proper subclasses.

On axiomatizability and decidability of universal theories of hereditary classes of matroids

Two hereditary classes of matroids – the class of matroids of rank doesn’t exceeding a fixed positive integer k and the class of matroids of finite rank – are studied by means of the model theory. The problems of axiomatizability of these two classes of matroids as structures and the problems of algorithmic decidability of universal theories of these classes are considered. It is shown that the first class is finitely axiomatizable whereas the second one is nonaxiomatizable. Decidability of the universal theories of the both classes is proved.

Graphs without large bicliques and well-quasi-orderability by the induced subgraph relation

Recently, Daligault, Rao and Thomass\'e asked in [3] if every hereditary class which is well-quasi-ordered by the induced subgraph relation is of bounded clique-width. There are two reasons why this questions is interesting. First, it connects two seemingly unrelated notions. Second, if the question is answered affirmatively, this will have a strong algorithmic consequence. In particular, this will mean (through the use of Courcelle theorem [2]), that any problem definable in Monadic Second Order Logic can be solved in a polynomial time on any class well-quasi-ordered by the induced subgraph relation. In the present paper, we answer this question affirmatively for graphs without large bicliques. Thus the above algorithmic consequence is true, for example, for classes of graphs of bounded degree.

Counting Perfect Matchings and the Switch Chain

We examine the problem of exactly or approximately counting all perfect matchings in hereditary classes of nonbipartite graphs. In particular, we consider the switch Markov chain of Diaconis, Graham, and Holmes. We determine the largest hereditary class for which the chain is ergodic, and define a large new hereditary class of graphs for which it is rapidly mixing. We go on to show that the chain has exponential mixing time for a slightly larger class. We also examine the question of ergodicity of the switch chain in an arbitrary graph. Finally, we give exact counting algorithms for three classes.

Price of Connectivity for the Vertex Cover Problem and the Dominating Set Problem: Conjectures and Investigation of Critical Graphs

The vertex cover problem and the dominating set problem are two well-known problems in graph theory. Their goal is to find the minimum size of a vertex subset satisfying some properties. Both hold a connected version, which imposes that the vertex subset must induce a connected component. To study the interdependence between the connected version and the original version of a problem, the Price of Connectivity ( $$PoC$$ ) was introduced by Cardinal and Levy (Theor Comput Sci 411(26–28):2581–2590, 2010) and Levy (Approximation algorithms for covering problems in dense graphs. Ph.D. thesis, Universite libre de Bruxelles, Brussels, 2009) as the ratio between invariants from the connected version and the original version of the problem. Camby et al. (Discret Math Theor Comput Sci 16:207–224, 2014) for the vertex cover problem, Camby and Schaudt (Discret Appl Math 177:53–59, 2014) for the dominating set problem characterized some classes of $$PoC$$ -Near-Perfect graphs, hereditary classes of graphs in which the Price of Connectivity is bounded by a fixed constant. Moreover, only for the vertex cover problem, Camby et al. (2014) introduced the notion of critical graphs, graphs that can appear in the list of forbidden induced subgraphs characterization. By definition, the Price of Connectivity of a critical graph is strictly greater than that of any proper induced subgraph. In this paper, we prove that for the vertex cover problem, every critical graph is either isomorphic to a cycle on 5 vertices or bipartite. To go further in the previous studies, we also present conjectures on $$PoC$$ -Near-Perfect graphs and critical graphs with the help of the computer software GraphsInGraphs (Camby and Caporossi in Studying graphs and their induced subgraphs with the computer: GraphsInGraphs. Cahiers du GERAD G-2016-10, 2016). Moreover, for the dominating set problem, we investigate critical trees and we show that every minimum dominating set of a critical graph is independent.

Counting Edge-injective Homomorphisms and Matchings on Restricted Graph Classes

We consider the #W[1]-hard problem of counting all matchings with exactly k edges in a given input graph G; we prove that it remains #W[1]-hard on graphs G that are line graphs or bipartite graphs with degree 2 on one side. In our proofs, we use that k-matchings in line graphs can be equivalently viewed as edge-injective homomorphisms from the disjoint union of k length-2 paths into (arbitrary) host graphs. Here, a homomorphism from H to G is edge-injective if it maps any two distinct edges of H to distinct edges in G. We show that edge-injective homomorphisms from a pattern graph H can be counted in polynomial time if H has bounded vertex-cover number after removing isolated edges. For hereditary classes $\mathcal {H}$ of pattern graphs, we complement this result: If the graphs in $\mathcal {H}$ have unbounded vertex-cover number even after deleting isolated edges, then counting edge-injective homomorphisms with patterns from $\mathcal {H}$ is #W[1]-hard. Our proofs rely on an edge-colored variant of Holant problems and a delicate interpolation argument; both may be of independent interest.

On the Complexity of the Vertex 3-Coloring Problem for the Hereditary Graph Classes With Forbidden Subgraphs of Small Size

The 3-coloring problem for a given graph consists in verifying whether it is possible to divide the vertex set of the graph into three subsets of pairwise nonadjacent vertices. A complete complexity classification is known for this problem for the hereditary classes defined by triples of forbidden induced subgraphs, each on at most 5 vertices. In this article, the quadruples of forbidden induced subgraphs is under consideration, each on atmost 5 vertices. For all but three corresponding hereditary classes, the computational status of the 3-coloring problem is determined. Considering two of the remaining three classes, we prove their polynomial equivalence and polynomial reducibility to the third class.

New forms of strong weakly $\mu$-compact in terms of hereditary classes

New forms of strong weakly $\mu$-compact in terms of hereditary classes

Chromatic bounds for some classes of [formula omitted]-free graphs

A hereditary class G of graphs is χ-bounded if there is a χ-binding function, say f such that χ(G)≤f(ω(G)), for every G∈G, where χ(G) (ω(G)) denotes the chromatic (clique) number of G. It is known that for every 2K2-free graph G, χ(G)≤ω(G)+12, and the class of (2K2,3K1)-free graphs does not admit a linear χ-binding function. In this paper, we are interested in classes of 2K2-free graphs that admit a linear χ-binding function. We show that the class of (2K2,H)-free graphs, where H∈{K1+P4,K1+C4,P2∪P3¯,HVN,K5−e,K5} admits a linear χ-binding function. Also, we show that some superclasses of 2K2-free graphs are χ-bounded.