### Book chapter

## Pq-König Extended Forests and Cycles

A graph is König for a q-path if every its induced subgraph has the following property. The maximum number of pairwise vertex-disjoint induced paths each on q vertices is equal to the minimum number of vertices, such that removing all the vertices produces a graph having no an induced path on q vertices. In this paper, for every q>4, we describe all Konig graphs for a q-path obtained from forests and simple sycles by replacing some vertices into graphs not containing induced paths on q vertices.

### In book

For a graph property X, let X_{n} be the set of graphs with the vertex set {1, . . . , n} that satisfy the property X. A property X is called factorial if X is hereditary (i. e. closed under taking induced subgraphs) and n^{c1n} ≤ X ≤ n^{c2n} for some positive constants c_{1} and c_{2}. A graph G is a *quasi-line* if for every vertex v, the set of neighbors of v can be expressed as a union of two cliques. In the present paper we identify almost all factorial subclasses of quasi-line graphs defined by one forbidden induced subgraph. We use these new results to prove that the class Free(K_{1,3},W_{4}) is factorial, which improves on a result of Lozin, Mayhill and Zamaraev [8].

For a graph property X, let X_{n} be the number of graphs with vertex set {1, . . . , 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^{c1n} ≤ X_{n} ≤ n^{c2n} for some constants c_{1} and c_{2}. Hereditary properties with speed slower than factorial are surprisingly well structured. The situation with factorial properties is more complicated and less explored. Only the properties with speeds up to the Bell number are well studied and well behaved. To better understand the behavior of factorial properties with faster speeds we introduce a structural tool called locally bounded coverings and show that a variety of graph properties can be described by means of this tool.

We consider a monopolistic firm that sells seasonal goods. The firm seeks the minimum of the total advertising expenditure during the selling period, given that some previously defined levels of goodwill and sales have to be reached at the end of the period. The only control allowed is on advertising while goodwill and sales levels are considered as state variables. More precisely we consider a linear optimal control problem for which the general position condition does not hold so that the application of Pontryagin's Maximum Principle may not be useful to determine a solution. Therefore the dual of the problem is studied and solved. Moreover, a necessary and sufficient condition for the feasibility of the primal problem is determined.

We characterize the graphs whose induced subgraphs all have the following property: The maximum number of induced 4-paths is equal to the minimum cardinality of the set of vertices such that every induced 4-path contains at least one of them. In this chapter we describe all such graphs obtained from simple cycles by replacing some vertices with cographs.

In the present paper the game theory is applied to an important open question in economics: providing microfoundations for often-used types of production function. Simple differential games of bargaining are proposed to model a behavior of workers and capital-owners in processes of formation of a set of admissible factor prices or participants’ weights (moral-ethical assessments). These games result, correspondingly, in a factor price curve and a weight curve – structures dual to production function. Ultimately, under constant bargaining powers of the participants, the Cobb-Douglas production function is received.

For a graph property X, let X_{n} be the number of graphs with vertex set {1, . . . , 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^{c1n} ≤ X_{n} ≤ n^{c2n} for some positive constants c_{1} and c_{2}. Hereditary properties with speed slower than factorial are surprisingly well structured. The situation with factorial properties is more complicated and less explored. To better understand the structure of factorial properties we look for minimal superfactorial ones. In [J.P. Spinrad, Nonredundant 1’s in *Γ*-free matrices, SIAM J. Discrete Math. 8 (1995) 251–257], Spinrad showed that the number of n-vertex chordal bipartite graphs is 2^{Θ(n log2n)}, which means that this class is superfactorial. On the other hand, all subclasses of chordal bipartite graphs that have been studied in the literature, such as forest, bipartite permutation, bipartite distance-hereditary or convex graphs, are factorial. In this paper, we study more hereditary subclasses of chordal bipartite graphs and reveal both factorial and superfactorial members in this family. The latter fact shows that the class of chordal bipartite graphs is not a minimal superfactorial one. Finding minimal superfactorial classes in this family remains a challenging open question.