Yury L. Orlovich

Hamiltonian properties of triangular grid graphs

A triangular grid graph is a finite induced subgraph of the infinite graph associated with the two-dimensional triangular grid. In 2000, Reay and Zamfirescu showed that all 2-connected, linearly-convex triangular grid graphs (with the exception of one of them) are hamiltonian. The only exception is a graph D which is the linearly-convex hull of the Star of David. We extend this result to a wider class of locally connected triangular grid graphs. Namely, we prove that all connected, locally connected triangular grid graphs (with the same exception of graph D) are hamiltonian. Moreover, we present a sufficient condition for a connected graph to be fully cycle extendable. We also show that the problem Hamiltonian Cycle is NP-complete for triangular grid graphs.

The complexity of dissociation set problems in graphs

A subset of vertices in a graph is called a dissociation set if it induces a subgraph with a vertex degree of at most 1. The maximum dissociation set problem, i.e., the problem of finding a dissociation set of maximum size in a given graph is known to be NP-hard for bipartite graphs. We show that the maximum dissociation set problem is NP-hard for planar line graphs of planar bipartite graphs. In addition, we describe several polynomially solvable cases for the problem under consideration. One of them deals with the subclass of the so-called chair-free graphs. Furthermore, the related problem of finding a maximal (by inclusion) dissociation set of minimum size in a given graph is studied, and NP-hardness results for this problem, namely for weakly chordal and bipartite graphs, are derived. Finally, we provide inapproximability results for the dissociation set problems mentioned above.

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.

Hamiltonian properties of locally connected graphs with bounded vertex degree

Abstract We consider the existence of Hamiltonian cycles for the locally connected graphs with a bounded vertex degree. For a graph G , let Δ ( G ) and δ ( G ) denote the maximum and minimum vertex degrees, respectively. We explicitly describe all connected, locally connected graphs with Δ ( G ) ⩽ 4 . We show that every connected, locally connected graph with Δ ( G ) = 5 and δ ( G ) ⩾ 3 is fully cycle extendable which extends the results of Kikust [P.B. Kikust, The existence of the Hamiltonian circuit in a regular graph of degree 5, Latvian Math. Annual 16 (1975) 33–38] and Hendry [G.R.T. Hendry, A strengthening of Kikust’s theorem, J. Graph Theory 13 (1989) 257–260] on full cycle extendability of the connected, locally connected graphs with the maximum vertex degree bounded by 5. Furthermore, we prove that problem Hamilton Cycle for the locally connected graphs with Δ ( G ) ⩽ 7 is NP-complete.

On the inapproximability of independent domination in 2P3-free perfect graphs

We consider the complexity of approximation for the Independent Dominating Set problem in 2P3-free graphs, i.e., graphs that do not contain two disjoint copies of the chordless path on three vertices as an induced subgraph. We show that, if P≠NP, the problem cannot be approximated for 2P3-free graphs in polynomial time within a factor of n1-e for any constant e>0, where n is the number of vertices in the graph. Moreover, we show that the result holds even if the 2P3-free graph is restricted to being weakly chordal (and thereby perfect).

Independent Domination in Triangle Graphs

Abstract We study the complexity of INDEPENDENT DOMINATION, a well-known algorithmical problem, for triangle graphs, i.e., graphs G satisfying the following triangle condition: for every maximal independent set I in G and every edge uv in G − I , there is a vertex w ∈ I such that { u , v , w } induces a triangle in G. We show that INDEPENDENT DOMINATION within triangle graphs is closely connected with the general STABLE MAX-CUT problem. However, the INDEPENDENT DOMINATION problem is NP-complete for K1,4-free triangle graphs. Finally, we investigate some natural invariants related to independent domination from the algorithmical point of view and apply our results to triangle graphs.

CYCLIC PROPERTIES OF TRIANGULAR GRID GRAPHS

Abstract It is known that all 2-connected, linearly convex triangular grid graphs, with only one exception, are hamiltonian (Reay and Zamfirescu, 2000). In the paper, it is shown that this result holds for a wider class of connected, locally connected triangular grid graphs and, with more exceptions, even for some general class of graphs. It is also shown that the HAMILTONIAN CYCLE problem is NP-complete for triangular grid graphs.

Graphs with maximal induced matchings of the same size

Abstract A graph is well-indumatched if all its maximal induced matchings are of the same size. We first prove that recognizing whether a graph is well-indumatched is a co-NP-complete problem even for ( 2 P 5 , K 1 , 5 ) -free graphs. We then show that decision problems Independent Dominating Set , Independent Set , and Dominating Set are NP-complete for the class of well-indumatched graphs. We also show that this class is a co-indumatching hereditary class, i.e., it is closed under deleting the end-vertices of an induced matching along with their neighborhoods, and we characterize well-indumatched graphs in terms of forbidden co-indumatching subgraphs. We prove that recognizing a co-indumatching subgraph is an NP-complete problem. We introduce a perfectly well-indumatched graph, in which every induced subgraph is well-indumatched, and characterize the class of these graphs in terms of forbidden induced subgraphs. Finally, we show that the weighted versions of problems Independent Dominating Set and Independent Set can be solved in polynomial time for perfectly well-indumatched graphs, but problem Dominating Set is NP-complete.

Squares of Intersection Graphs and Induced Matchings

Abstract We consider the squares of intersection graphs of hypergraphs and simple graphs. The Berge-type characterization for squares of intersection graphs of hypergraphs is obtained. The analogue of Whitney theorem for squares of intersection graphs of trees is proved. Using the connection between induced matchings of graph and independent sets of square of line graph, new polynomial solvable cases for the weighted induced matching problem are obtained.

Graphs with Maximal Induced Matchings of the Same Size

Abstract A graph is well-indumatched if all its maximal induced matchings are of the same size. We first prove that recognizing the class WIM of well-indumatched graphs is a co-NP-complete problem even for (2 P 5 , K 1,5 )-free graphs. We then show that the well-known decision problems such as Independent Dominating Set, Independent Set, and Dominating Set are NP-complete for well-indumatched graphs. We also show that WIM is a co-indumatching hereditary class and characterize well-indumatched graphs in terms of forbidden co-indumatching subgraphs. However, we prove that recognizing co-indumatching subgraphs is an NP-complete problem. A graph G is perfectly well-indumatched if every induced subgraph of G is well-indumatched. We characterize the class of perfectly well-indumatched graphs in terms of forbidden induced subgraphs. Finally, we show that both Independent Dominating Set and Independent Set can be solved in polynomial time for perfectly well-indumatched graphs, even in their weighted versions, but Dominating Set is still NP-complete.