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Vadim Lozin

Publications and source records attributed to Vadim Lozin.

29 records · Page 2Linked to original sources
arXiv2017

Perfect Edge Domination: Hard and Solvable Cases

Let $G$ be an undirected graph. An edge of $G$ dominates itself and all edges adjacent to it. A subset $E'$ of edges of $G$ is an edge dominating set of $G$, if every edge of the graph is dominated by some edge of $E'$. We say that $E'$ is a perfect edge dominating set of $G$, if every edge not in $E'$ is dominated by exactly one edge of $E'$. The perfect edge dominating problem is to determine a least cardinality perfect edge dominating set of $G$. For this problem, we describe two NP-completeness proofs, for the classes of claw-free graphs of degree at most 3, and for bounded degree graphs, of maximum degree at most $d \geq 3$ and large girth. In contrast, we prove that the problem admits an $O(n)$ time solution, for cubic claw-free graphs. In addition, we prove a complexity dichotomy theorem for the perfect edge domination problem, based on the results described in the paper. Finally, we describe a linear time algorithm for finding a minimum weight perfect edge dominating set of a $P_5$-free graph. The algorithm is robust, in the sense that, given an arbitrary graph $G$, either it computes a minimum weight perfect edge dominating set of $G$, or it exhibits an induced subgraph of $G$, isomorphic to a $P_5$.

cs.DM↗
arXiv2017

More results on weighted independent domination

Weighted independent domination is an NP-hard graph problem, which remains computationally intractable in many restricted graph classes. In particular, the problem is NP-hard in the classes of sat-graphs and chordal graphs. We strengthen these results by showing that the problem is NP-hard in a proper subclass of the intersection of sat-graphs and chordal graphs. On the other hand, we identify two new classes of graphs where the problem admits polynomial-time solutions.

cs.DM↗
arXiv2016

Upper Domination: towards a dichotomy through boundary properties

An upper dominating set in a graph is a minimal (with respect to set inclusion) dominating set of maximum cardinality. The problem of finding an upper dominating set is generally NP-hard. We study the complexity of this problem in classes of graphs defined by finitely many forbidden induced subgraphs and conjecture that the problem admits a dichotomy in this family, i.e. it is either NP-hard or polynomial-time solvable for each class in the family. A helpful tool to study the complexity of an algorithmic problem on finitely defined classes of graphs is the notion of boundary classes. However, none of such classes has been identified so far for the upper dominating set problem. In the present paper, we discover the first boundary class for this problem and prove the dichotomy for classes defined by a single forbidden induced subgraph.

cs.DM↗
arXiv2016

The structure and the number of $P_7$-free bipartite graphs

We show that the number of labelled $P_7$-free bipartite graphs with $n$ vertices grows as $n^{Θ(n)}$. This resolves an open problem posed by Allen [P. Allen, Forbidden induced bipartite graphs. J. Graph Theory 60 (2009), no. 3, 219--241.], and completes the description of speeds of monogenic classes of bipartite graphs. Our solution is based on a new decomposition scheme of bipartite graphs, which is of independent interest.

math.CO↗
arXiv2015

Dominating induced matchings in graphs containing no long claw

An induced matching $M$ in a graph $G$ is dominating if every edge not in $M$ shares exactly one vertex with an edge in $M$. The dominating induced matching problem (also known as efficient edge domination) asks whether a graph $G$ contains a dominating induced matching. This problem is generally NP-complete, but polynomial-time solvable for graphs with some special properties. In particular, it is solvable in polynomial time for claw-free graphs. In the present paper, we study this problem for graphs containing no long claw, i.e. no induced subgraph obtained from the claw by subdividing each of its edges exactly once. To solve the problem in this class, we reduce it to the following question: given a graph $G$ and a subset of its vertices, does $G$ contain a matching saturating all vertices of the subset? We show that this question can be answered in polynomial time, thus providing a polynomial-time algorithm to solve the dominating induced matching problem for graphs containing no long claw.

cs.DM↗
arXiv2015

Well-Quasi-Order for Permutation Graphs Omitting a Path and a Clique

We consider well-quasi-order for classes of permutation graphs which omit both a path and a clique. Our principle result is that the class of permutation graphs omitting $P_5$ and a clique of any size is well-quasi-ordered. This is proved by giving a structural decomposition of the corresponding permutations. We also exhibit three infinite antichains to show that the classes of permutation graphs omitting $\{P_6,K_6\}$, $\{P_7,K_5\}$, and $\{P_8,K_4\}$ are not well-quasi-ordered.

math.CO↗
arXiv2015

Well-quasi-ordering does not imply bounded clique-width

We present a hereditary class of graphs of unbounded clique-width which is well-quasi-ordered by the induced subgraph relation. This result provides a negative answer to the question asked by Daligault, Rao and Thomassé in ("Well-quasi-order of relabel functions", Order, 27(3):301--315, 2010).

cs.DM↗
arXiv2014

Implicit Representations and Factorial Properties of Graphs

The idea of implicit representation of graphs was introduced in [S. Kannan, M. Naor, S. Rudich, Implicit representation of graphs, SIAM J. Discrete Mathematics, 5 (1992) 596--603] and can be defined as follows. A representation of an $n$-vertex graph $G$ is said to be implicit if it assigns to each vertex of $G$ a binary code of length $O(\log n)$ so that the adjacency of two vertices is a function of their codes. Since an implicit representation of an $n$-vertex graph uses $O(n\log n)$ bits, any class of graphs admitting such a representation contains $2^{O(n\log n)}$ labelled graphs with $n$ vertices. In the terminology of [J. Balogh, B. Bollobás, D. Weinreich, The speed of hereditary properties of graphs, J. Combin. Theory B 79 (2000) 131--156] such classes have at most factorial speed of growth. In this terminology, the implicit graph conjecture can be stated as follows: every class with at most factorial speed of growth which is hereditary admits an implicit representation. The question of deciding whether a given hereditary class has at most factorial speed of growth is far from being trivial. In the present paper, we introduce a number of tools simplifying this question. Some of them can be used to obtain a stronger conclusion on the existence of an implicit representation. We apply our tools to reveal new hereditary classes with the factorial speed of growth. For many of them we show the existence of an implicit representation.

math.CO↗
arXiv2014

New results on word-representable graphs

A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $(x,y)\in E$ for each $x\neq y$. The set of word-representable graphs generalizes several important and well-studied graph families, such as circle graphs, comparability graphs, 3-colorable graphs, graphs of vertex degree at most 3, etc. By answering an open question from [M. Halldorsson, S. Kitaev and A. Pyatkin, Alternation graphs, Lect. Notes Comput. Sci. 6986 (2011) 191--202. Proceedings of the 37th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2011, Tepla Monastery, Czech Republic, June 21-24, 2011.], in the present paper we show that not all graphs of vertex degree at most 4 are word-representable. Combining this result with some previously known facts, we derive that the number of $n$-vertex word-representable graphs is $2^{\frac{n^2}{3}+o(n^2)}$.

math.CO↗
arXiv2007

Deciding k-colourability of $P_5$-free graphs in polynomial time

The problem of computing the chromatic number of a $P_5$-free graph is known to be NP-hard. In contrast to this negative result, we show that determining whether or not a $P_5$-free graph admits a $k$-colouring, for each fixed number of colours $k$, can be done in polynomial time. If such a colouring exists, our algorithm produces it.

cs.DS↗