Igor E. Zverovich

Characterizations of closed classes of Boolean functions in terms of forbidden subfunctions and Post classes

We characterize Post classes of Boolean functions (also known as clones) in terms of forbidden subfunctions that allows one to give a comparably short proof of the classical Post theorem.

An induced subgraph characterization of domination perfect graphs

Let γ(G) ι(G) be the domination number and independent domination number of a graph (G), respectively. A graph (G) is called domination perfect if γ(H) = ι(H), for every induced subgraph H of (G). There are many results giving a partial characterization of domination perfect graphs. In this paper, we present a finite induced subgraph characterization of the entire class of domination perfect graphs. The list of forbidden subgraphs in the charcterization consists of 17 minimal domination imperfect graphs. Moreover, the dominating set and independent dominating set problems are shown to be both NP‐complete on some classes of graphs.

Contributions to the theory of graphic sequences

Abstract In this article we present a new version of the Erdos-Gallai theorem concerning graphicness of the degree sequences. The best conditions of all known on the reduction of the number of Erdos-Gallai inequalities are given. Moreover, we prove a criterion of the bipartite graphicness and give a sufficient condition for a sequence to be graphic which does not require checking of any Erdos-Gallai inequality.

A semi-induced subgraph characterization of upper domination perfect graphs

Using a variation of Thomassens admissible triples technique, we give an alternative proof that every strongly 2-arc-connected directed graph with two or more vertices contains a directed cycle that has at least two chords, while at the same time establishing a more general result.

Approximability results for the maximum and minimum maximal induced matching problems

An induced matchingM of a graph G is a set of pairwise non-adjacent edges such that their end-vertices induce a 1-regular subgraph. An induced matching M is maximal if no other induced matching contains M. The Minimum Maximal Induced Matching problem asks for a minimum maximal induced matching in a given graph. The problem is known to be NP-complete. Here we show that, if P NP, for any @e>0, this problem cannot be approximated within a factor of n^1^-^@e in polynomial time, where n denotes the number of vertices in the input graph. The result holds even if the graph in question is restricted to being bipartite. For the related problem of finding an induced matching of maximum size (Maximum Induced Matching), it is shown that, if P NP, for any @e>0, the problem cannot be approximated within a factor of n^1^/^2^-^@e in polynomial time. Moreover, we show that Maximum Induced Matching is NP-complete for planar line graphs of planar bipartite graphs.

Perfect connected-dominant graphs

If D is a dominating set and the induced subgraph G(D) is connected, then D is a connected dominating set. The minimum size of a connected dominating set in G is called connected domination number ∞c(G) of G. A graph G is called a perfect connected-dominant graph if ∞(H) = ∞c(H) for each connected induced subgraph H of G. We prove that a graph is a perfect connected-dominant graph if and only if it contains no induced path P5 and induced cycle C5.

Extension of hereditary classes with substitutions

Let G and H be graphs. A substitution of H in G instead of a vertex ν ∈ V(G) is the graph G(ν → H), which consists of disjoint union of H and G - ν with the additional edge-set {xy: x ∈ V(H), y ∈ NG(ν)}.For a hereditary class of graphs P, the substitutional closure of P is defined as the class P* consisting of all graphs which can be obtained from graphs in P by repeated substitutions.Let P be an arbitrary hereditary class for which a characterization in terms of forbidden induced subgraphs is known. We propose a method of constructing forbidden induced subgraphs for P*.

A new kind of graph coloring

A proper k-coloring C1, C2,...,Ck of a graph G is called strong if, for every vertex u ∈ V(G), there exists an index i ∈ {1, 2,...,k} such that u is adjacent to every vertex of Ci. We consider classes SCOLOR(k) of strongly k-colorable graphs and show that the recognition problem of SCOLOR(k) is NP-complete for every k ≥ 4, but it is polynomial-time solvable for k = 3. We give a characterization of SCOLOR(3) in terms of forbidden induced subgraphs. Finally, we solve the problem of uniqueness of a strong 3-coloring.

A characterization of domination perfect graphs

Let γ(G) and i(G) be the domination number and independent domination number of a graph G, respectively. Sumner and Moore [8] define a graph G to be domination perfect if γ(H) = i(H), for every induced subgraph H of G. In this article, we give a finite forbidden induced subgraph characterization of domination perfect graphs. Bollobas and Cockayne [4] proved an inequality relating γ(G) and i(G) for the class of K1,k ‐free graphs. It is shown that the same inequality holds for a wider class of graphs. Copyright

r-Bounded k-complete bipartite bihypergraphs and generalized split graphs

Let P and Q be hereditary classes of graphs. Suppose that the stability number α(H) is bounded above for all H ∈ P, and the clique number ω(H) is bounded above for all H ∈ Q. An ordered partition A∪B=V(G) is called a Ramseian (P,Q)-partition if G(A)∈ P and G(B)∈Q where G(X) denotes the subgraph of G induced by X. Let R(P,Q) be the set of all graphs which have a Ramseian (P,Q)-partition. It is proved that if both P and Q have finite forbidden induced subgraph (FIS) characterizations then R(P,Q) also has such a characterization. In particular, every class of (α,β)-polar graphs (which are generalizations of split graphs) has a finite FIS-characterization.For the proof we use the following model. Let H0 and H1 be hypergraphs with the same vertex set V. The ordered pair H=(H0,H1) is called a bihypergraph. A bihypergraph H=(H0,H1) is called bipartite if there is an ordered partition V0 ∪ V1 = V(H) such that Vi is stable in Hi, i=0, 1. If the maximum cardinality of hyperedges in H is at most r and every k-subset of V(H) contains at least one hyperedge then H∈C(k,r). We prove that there exists a finite number of minimal non-bipartite bihypergraphs in C(k,r) (when k and r are fixed).