Martin Milanič

Recent developments on graphs of bounded clique-width

Whether the clique-width of graphs in a certain class of graphs is bounded or not, is an important question from an algorithmic point of view, as many problems that are NP-hard in general admit polynomial-time solutions when restricted to graphs of bounded clique-width. Over the last few years, many classes of graphs have been shown to have bounded clique-width. For many others, this parameter has been proved to be unbounded. This paper provides a survey of recent results addressing this problem.

Shortest paths between shortest paths

a b s t r a c t We study the following problem on reconfiguring shortest paths in graphs: Given two shortest s-t paths, what is the minimum number of steps required to transform one into the other, where each intermediate path must also be a shortest s-t path and must differ from the previous one by only one vertex. We prove that the shortest reconfiguration sequence can be exponential in the size of the graph and that it is NP-hard to compute the shortest reconfiguration sequence even when we know that the sequence has polynomial length.

Latency-Bounded Target Set Selection in Social Networks

We study variants of the Target Set Selection problem, first proposed by Kempe et al. In our scenario one is given a graph G = (V,E), integer values t(v) for each vertex v, and the objective is to determine a small set of vertices (target set) that activates a given number (or a given subset) of vertices of G within a prescribed number of rounds. The activation process in G proceeds as follows: initially, at round 0, all vertices in the target set are activated; subsequently at each round r ≥ 1 every vertex of G becomes activated if at least t(v) of its neighbors are active by round r − 1. It is known that the problem of finding a minimum cardinality Target Set that eventually activates the whole graph G is hard to approximate to a factor better than \(O(2^{\log^{1-\epsilon }|V|})\). In this paper we give exact polynomial time algorithms to find minimum cardinality Target Sets in graphs of bounded clique-width, and exact linear time algorithms for trees.

Graphs of separability at most 2

We introduce graphs of separability at mostk as graphs in which every two non-adjacent vertices are separated by a set of at most k other vertices. Graphs of separability at most k arise in connection with the Parsimony Haplotyping problem from computational biology. For k@?{0,1}, the only connected graphs of separability at most k are complete graphs and block graphs, respectively. For k>=3, graphs of separability at most k form a rich class of graphs containing all graphs of maximum degree k. We prove several characterizations of graphs of separability at most 2, which generalize complete graphs, cycles and trees. The main result is that every connected graph of separability at most 2 can be constructed from complete graphs and cycles by pasting along vertices or edges, and vice versa, every graph constructed this way is of separability at most 2. The structure theorem has nice algorithmic implications-some of which cannot be extended to graphs of higher separability-however certain optimization problems remain intractable on graphs of separability 2. We then characterize graphs of separability at most 2 in terms of minimal forbidden induced subgraphs and minimal forbidden induced minors. Finally, we discuss the possibilities of extending these results to graphs of higher separability.

Set graphs. I. Hereditarily finite sets and extensional acyclic orientations

A graph G is said to be a set graph if it admits an acyclic orientation which is also extensional, in the sense that the out-neighborhoods of its vertices are pairwise distinct. Equivalently, a set graph is the underlying graph of the digraph representation of a hereditarily finite set. In this paper, we initiate the study of set graphs. On the one hand, we identify several necessary conditions that every set graph must satisfy. On the other hand, we show that set graphs form a rich class of graphs containing all connected claw-free graphs and all graphs with a Hamiltonian path. In the case of claw-free graphs, we provide a polynomial-time algorithm for finding an extensional acyclic orientation. Inspired by manipulations of hereditarily finite sets, we give simple proofs of two well-known results about claw-free graphs. We give a complete characterization of unicyclic set graphs, and point out two NP-complete problems closely related to the problem of recognizing set graphs. Finally, we argue that these three problems are solvable in linear time on graphs of bounded treewidth.

Hereditary Efficiently Dominatable Graphs

An efficient dominating set (or perfect code) in a graph is a set of vertices the closed neighborhoods of which partition the graphs vertex set. We introduce graphs that are hereditary efficiently dominatable in that sense that every induced subgraph of the graph contains an efficient dominating set. We prove a decomposition theorem for (bull, fork, C4)-free graphs, based on which we characterize, in terms of forbidden induced subgraphs, the class of hereditary efficiently dominatable graphs. We also give a decomposition theorem for hereditary efficiently dominatable graphs and examine some algorithmic aspects of such graphs. In particular, we give a polynomial time algorithm for finding an efficient dominating set (if one exists) in a class of graphs properly containing the class of hereditary efficiently dominatable graphs by reducing the problem to the maximum weight independent set problem in claw-free graphs.

Computing square roots of trivially perfect and threshold graphs

A graph H is a square root of a graph G if two vertices are adjacent in G if and only if they are at distance one or two in H. Computing a square root of a given graph is NP-hard, even when the input graph is restricted to be chordal. In this paper, we show that computing a square root can be done in linear time for a well-known subclass of chordal graphs, the class of trivially perfect graphs. This result is obtained by developing a structural characterization of graphs that have a split square root. We also develop linear time algorithms for determining whether a threshold graph given either by a degree sequence or by a separating structure has a square root.

Shortest paths between shortest paths and independent sets

We study problems of reconfiguration of shortest paths in graphs. We prove that the shortest reconfiguration sequence can be exponential in the size of the graph and that it is NP-hard to compute the shortest reconfiguration sequence even when we know that the sequence has polynomial length. Moreover, we also study reconfiguration of independent sets in three different models and analyze relationships between these models, observing that shortest path reconfiguration is a special case of independent set reconfiguration in perfect graphs, under any of the three models. Finally, we give polynomial results for restricted classes of graphs (even-hole-free and P4-free graphs).

On the maximum independent set problem in subclasses of planar graphs

The maximum independent set problem is known to be NP-hard in the class of planar graphs. In the present paper, we study its complexity in hereditary subclasses of planar graphs. In particular, by combining various techniques, we show that the problem is polynomially solvable in the class of S1,2,k-free planar graphs, generalizing several previously known results. S1,2,k is the graph consisting of three induced paths of lengths 1, 2 and k, with a common initial vertex.

Polynomial-time algorithms for weighted efficient domination problems in AT-free graphs and dually chordal graphs

An efficient dominating set (or perfect code) in a graph is a set of vertices the closed neighborhoods of which partition the vertex set of the graph. The minimum weight efficient domination problem is the problem of finding an efficient dominating set of minimum weight in a given vertex-weighted graph; the maximum weight efficient domination problem is defined similarly. We develop a framework for solving the weighted efficient domination problems based on a reduction to the maximum weight independent set problem in the square of the input graph. Using this approach, we improve on several previous results from the literature by deriving polynomial-time algorithms for the weighted efficient domination problems in the classes of dually chordal and AT-free graphs. In particular, this answers a question by Lu and Tang regarding the complexity of the minimum weight efficient domination problem in strongly chordal graphs. A framework for solving the minimum weighted efficient domination (Min-WED) problem is developed.The problem is reduced to the maximum weight independent set problem in the square.The Min-WED problem is polynomial for AT-free graphs and dually chordal graphs.In particular, this answers a question by Lu and Tang.The approach also works for the maximization version of the problem.