A graph class is monotone if it is closed under taking subgraphs. A monotone class defined by finitely many obstructions has bounded treewidth if and only if one of the obstructions is a tripod, i.e. a disjoint union of subdivided claws and paths. This dichotomy also characterizes exactly those monotone graph classes for which many NP-hard graph problems admit polynomial-time algorithms. These dichotomies do not extend to the universe of all hereditary classes. This leads to the question of whether we can extend known dichotomies for monotone classes to larger families of hereditary classes. We answer this question affirmatively by considering the family of hereditary graph classes closed under self-intersection. This family is known to be located strictly between the monotone and hereditary classes. We prove a new structural characterization of graphs in self-intersection-closed classes excluding a tripod. In contrast to monotone classes excluding a tripod, these classes do not necessarily have bounded treewidth; in fact, they do not even need to be sparse. We use our characterization to give a complete dichotomy for Maximum Independent Set, and its weighted variant, on self-intersection-closed classes defined by finitely many obstructions: these problems are in P if the class excludes a tripod and NP-hard otherwise. Our dichotomy generalizes several known results on Maximum Independent Set in the literature. We also apply our characterization to obtain a dichotomy for Maximum Induced Matching on self-intersection-closed classes of bipartite graphs defined by finitely many obstructions, and for Satisfiability and Counting Satisfiability on self-intersection-closed classes of (bipartite) incidence graphs defined by finitely many obstructions. Finally, we use our characterization to obtain a dichotomy for boundedness of clique-width for self-intersection-closed classes of bipartite graphs defined by finitely many obstructions.

Abstract Lettericity is a graph parameter responsible for many attractive structural properties. In particular, graphs of bounded lettericity have bounded linear clique-width and they are well-quasi-ordered by induced subgraphs. The latter property implies that any hereditary class of graphs of bounded lettericity can be described by finitely many forbidden induced subgraphs. This, in turn, implies, in a non-constructive way, polynomial-time recognition of such classes. However, no constructive algorithms and no specific bounds on the size of forbidden graphs are available up to date. In the present paper, we develop an algorithm that recognizes n -vertex graphs of lettericity at most k in time $$f(k) \cdot n^3$$ and show that any minimal graph of lettericity more than k has at most $$2^{O(k^2\log k)}$$ vertices.

We study properties of graph classes that are closed under taking subclasses, such as boundedness of graph parameters or polynomial-time solvability of algorithmic problems. In the universe of minor-closed classes of graphs, any such property can be described by a set of minimal classes that do not possess the property, because the minor relation is a well-quasi-order. This, however, is not the case for the subgraph relation, implying that in the universe of monotone classes, which extends the family of minor-closed classes, the existence of minimal classes is not guaranteed. To overcome this difficulty, we employ the notion of boundary classes. Together with minimal classes they play a critical role for classes defined by finitely many forbidden subgraphs. In the present paper, we identify several levels in the hierarchy of monotone classes and describe respective critical classes. In particular, we show that a finitely-defined monotone class X has bounded chromatic number, degeneracy, functionality and admits an implicit representation if and only if X excludes a forest. We also show that X has bounded tree-, clique- and twin-width and admits polynomial-time solutions for a variety of algorithmic problems if and only if X excludes a tripod, i.e. a subcubic forest every connected component of which has at most one cubic vertex. The last result, however, does not apply to the Hamiltonian cycle problem. Towards identifying critical classes for this problem we determine complexity of the Hamiltonian cycle problem in some monotone classes.

The maximum weight independent set problem (WIS), which is known to be generally NP-hard, admits polynomial-time solutions when restricted to graphs in some special classes. In particular, due to the celebrated Edmonds' matching algorithm , WIS is solvable in polynomial time in the class of line graphs. This solution was extended to claw-free graphs and then further to fork-free graphs and to t claw-free graphs, where t claw is the graph consisting of t disjoint copies of the claw. The solution for t claw-free graphs was obtained by generalizing Farber's approach to solve the problem for t K 2 -free graphs. In the present paper, we elaborate this approach further to develop a polynomial-time algorithm to solve the problem in the class of fork+ t claw-free graphs, generalizing both fork-free graphs and t claw-free graphs, and in the class of P 5 + t claw-free graphs. We then apply the latter result to solve the more general problem of finding a d -regular induced subgraph of maximum weight in the class of P 5 + t P 3 -free graphs in polynomial time for any natural d and t , extending some of the previously known solutions.

Abstract. Given two [Formula: see text]-vertex graphs [Formula: see text] and [Formula: see text] of bounded treewidth, is there an [Formula: see text]-vertex graph [Formula: see text] of bounded treewidth having subgraphs isomorphic to [Formula: see text] and [Formula: see text]? Our main result is a negative answer to this question, in a strong sense: we show that the answer is no even if [Formula: see text] is a binary tree and [Formula: see text] is a ternary tree. We also provide an extensive study of cases where such “gluing” is possible. In particular, we prove that if [Formula: see text] has treewidth [Formula: see text] and [Formula: see text] has pathwidth [Formula: see text], then there is an [Formula: see text]-vertex graph of treewidth at most [Formula: see text] containing both [Formula: see text] and [Formula: see text] as subgraphs.

Abstract The class of quasi-chain graphs is an extension of the well-studied class of chain graphs. This latter class enjoys many nice and important properties, such as bounded clique-width, implicit representation, well-quasi-ordering by induced subgraphs, etc. The class of quasi-chain graphs is substantially more complex. In particular, this class is not well-quasi-ordered by induced subgraphs, and the clique-width is not bounded in it. In the present paper, we show that the universe of quasi-chain graphs is at least as complex as the universe of permutations by establishing a bijection between the class of all permutations and a subclass of quasi-chain graphs. This implies, in particular, that the induced subgraph isomorphism problem is NP-complete for quasi-chain graphs. On the other hand, we propose a decomposition theorem for quasi-chain graphs that implies an implicit representation for graphs in this class and efficient solutions for some algorithmic problems that are generally intractable.

The world of hereditary classes is rich and diverse and it contains a variety of classes of theoretical and practical importance. Thousands of results in the literature are devoted to individual classes and only a few of them analyse the universe of hereditary classes as a whole. To shift the analysis into a new level, in the present paper we exploit an approach, where we operate by infinite families of classes, rather than individual classes. Each family is associated with a graph parameter and is characterized by classes that are critical with respect to the parameter. In particular, we obtain a complete parametric description of the bottom of the lattice of hereditary classes and discuss a number of open questions related to this approach.

A set S of Boolean points is a specifying set for a threshold function f if the only threshold function consistent with f on S is f itself. The minimal cardinality of a specifying set for f is the specification number of f and it is never smaller than n+1 for a function with n relevant variables. In the present paper, we develop an inductive approach to describing the set of Boolean threshold functions with minimum specification number by means of operations that allow us to extend functions of n variables in this set to functions of n+1 variables.

We uncover a connection between two seemingly unrelated notions: lettericity, from structural graph theory, and geometric griddability, from the world of permutation patterns. Both of these notions capture important structural properties of their respective classes of objects. We prove that these notions are equivalent in the sense that a permutation class is geometrically griddable if and only if the corresponding class of inversion graphs has bounded lettericity.

We obtain a complete classification of graphs H for which the class of (triangle,H)-free graphs is well-quasi-ordered by the induced subgraph relation and an almost complete classification of graphs H for which the class of (triangle,H)-free graphs has bounded clique-width. In particular, we show that for these graph classes, well-quasi-orderability implies boundedness of clique-width. To obtain our results, we further refine a known method based on canonical decomposition. This leads to a new decomposition technique that is applicable to both notions, well-quasi-orderability and clique-width.

Clique-width is an important graph parameter due to its algorithmic and structural properties. A graph class is hereditary if it can be characterized by a (not necessarily finite) set ${\cal H}$ of forbidden induced subgraphs. We study the boundedness of clique-width of hereditary graph classes closed under complementation. First, we extend the known classification for the $|{\cal H}|=1$ case by classifying the boundedness of clique-width for every set ${\cal H}$ of self-complementary graphs. We then completely settle the $|{\cal H}|=2$ case. In particular, we determine one new class of $(H,\overline{H})$-free graphs of bounded clique-width (as a side effect, this leaves only five classes of $(H_1,H_2)$-free graphs, for which it is not known whether their clique-width is bounded). Once we have obtained the classification of the $|{\cal H}|=2$ case, we research the effect of forbidding self-complementary graphs on the boundedness of clique-width. Surprisingly, we show that for every set ${\cal F}$ of self-complementary graphs on at least five vertices, the classification of the boundedness of clique-width for $(\{H,\overline{H}\}\cup {\cal F})$-free graphs coincides with the one for the $|{\cal H}|=2$ case if and only if ${\cal F}$ does not include the bull.

An extremal point of a positive threshold Boolean function $f$ is either a maximal zero or a minimal one. It is known that if $f$ depends on all its variables, then the set of its extremal points completely specifies $f$ within the universe of threshold functions. However, in some cases, $f$ can be specified by a smaller set. The minimum number of points in such a set is the specification number of $f$. Hu (1965) showed that the specification number of a threshold function of $n$ variables is at least $n+1$. Anthony et al. (1995) proved that this bound is attained for nested functions and conjectured that for all other threshold functions the specification number is strictly greater than $n+1$. In the present paper, we resolve this conjecture negatively by exhibiting threshold Boolean functions of $n$ variables, which are non-nested and for which the specification number is $n+1$. On the other hand, we show that the set of extremal points satisfies the statement of the conjecture, i.e.~a positive threshold Boolean function depending on all its $n$ variables has $n+1$ extremal points if and only if it is nested. To prove this, we reveal an underlying structure of the set of extremal points.

Clique-width is an important graph parameter due to its algorithmic and structural properties. A graph class is hereditary if it can be characterized by a (not necessarily finite) set H of forbidden induced subgraphs. We initiate a systematic study into the boundedness of clique-width of hereditary graph classes closed under complementation. First, we extend the known classification for the |H|=1 case by classifying the boundedness of clique-width for every set H of self-complementary graphs. We then completely settle the |H|=2 case. In particular, we determine one new class of (H1, complement of H1)-free graphs of bounded clique-width (as a side effect, this leaves only six classes of (H1, H2)-free graphs, for which it is not known whether their clique-width is bounded). Once we have obtained the classification of the |H|=2 case, we research the effect of forbidding self-complementary graphs on the boundedness of clique-width. Surprisingly, we show that for a set F of self-complementary graphs on at least five vertices, the classification of the boundedness of clique-width for ({H1, complement of H1} + F)-free graphs coincides with the one for the |H|=2 case if and only if F does not include the bull (the only non-empty self-complementary graphs on fewer than five vertices are P_1 and P_4, and P_4-free graphs have clique-width at most 2). Finally, we discuss the consequences of our results for COLOURING.

The paper [J. Balogh, B. Bollobás, D. Weinreich, J. Combin. Theory Ser. B, 95 (2005), pp. 29--48] identifies a jump in the speed of hereditary graph properties to the Bell number $B_n$ and provides a partial characterization of the family of minimal classes whose speed is at least $B_n$. In the present paper, we give a complete characterization 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 question positively by showing that there exists an algorithm which, given a finite set $\mathcal{F}$ of graphs, decides whether the speed of the class of graphs containing no induced subgraphs from the set $\mathcal{F}$ is above or below the Bell number. For properties defined by infinitely many minimal forbidden induced subgraphs, the speed is known to be above the Bell number.

We present the first polynomial-time algorithm to solve the maximum weight independent set problem for apple-free graphs, which is a common generalization of several important classes where the problem can be solved efficiently, such as claw-free graphs, chordal graphs, and cographs. Our solution is based on a combination of two algorithmic techniques (modular decomposition and decomposition by clique separators) and a deep combinatorial analysis of the structure of apple-free graphs. Our algorithm is robust in the sense that it does not require the input graph G to be apple-free; the algorithm either finds an independent set of maximum weight in G or reports that G is not apple-free.

The band-, tree-, and clique-width are of primary importance in algorithmic graph theory due to the fact that many problems that are NP-hard for general graphs can be solved in polynomial time when restricted to graphs where one of these parameters is bounded. It is known that for any fixed $\Delta \geq 3$, all three parameters are unbounded for graphs with vertex degree at most $Delta$. In this paper, we distinguish representative subclasses of graphs with bounded vertex degree that have bounded band-, tree-, or clique-width. Our proofs are constructive and lead to efficient algorithms for a variety of NP-hard graph problems when restricted to those classes.

The 3-colorability problem is known to be NP-complete in the class of graphs with maximum degree four. On the other hand, due to the celebrated theorem of Brooks, the problem has a polynomial-time solution for graphs with maximum degree three. To make the complexity gap more precise, we study a family of intermediate graph classes between these two extremes and classify all of them according to the computational complexity of the problem. In particular, we generalize Brooks's theorem in the case of 3-colorability to a larger class by showing that every connected graph in that class is 3-colorable, unless it is a complete graph on four vertices.