Rodica Mihai

Pathwidth is NP-Hard for Weighted Trees

The pathwidth of a graph G is the minimum clique number of H minus one, over all interval supergraphs H of G . We prove in this paper that the pathwidth problem is NP-hard for particular subclasses of chordal graphs, and we deduce that the problem remains hard for weighted trees. We also discuss subclasses of chordal graphs for which the problem is polynomial.

Cutwidth of Split Graphs and Threshold Graphs

We give a linear-time algorithm to compute the cutwidth of threshold graphs, thereby resolving the computational complexity of cutwidth on this graph class. Threshold graphs are a well-studied subclass of interval graphs and of split graphs, both of which are unrelated subclasses of chordal graphs. To complement our result, we show that cutwidth is NP-complete on split graphs, and consequently also on chordal graphs. The cutwidth of interval graphs is still open, and only very few graph classes are known so far on which polynomial-time cutwidth algorithms exist. Thus we contribute to define the border between graph classes on which cutwidth is polynomially solvable and on which it remains NP-complete.

Mixed Search Number of Permutation Graphs

Search games in graphs have attracted significant attention in recent years, and they have applications in securing computer networks against viruses and intruders. Since graph searching is an NP-hard problem, polynomial-time algorithms have been given for solving it on various graph classes. Most of these algorithms concern computing the node search number of a graph, and only few such algorithms are known for computing the mixed search or edge search numbers of specific graph classes. In this paper we show that the mixed search number of permutation graphs can be computed in linear time, and we describe an algorithm for this purpose. In addition, we give a complete characterization of the edge search number of complete bipartite graphs.

Mixed search number and linear-width of interval and split graphs

We show that the mixed search number and the linear-width of interval graphs and of split graphs can be computed in linear time and in polynomial time, respectively.

Edge Search Number of Cographs in Linear Time

We give a linear-time algorithm for computing the edge search number of cographs, thereby proving that this problem can be solved in polynomial time on this graph class. With our result, the knowledge on graph searching of cographs is now complete: node, mixed, and edge search numbers of cographs can all be computed efficiently. Furthermore, we are one step closer to computing the edge search number of permutation graphs.

Edge search number of cographs

We give a linear-time algorithm for computing the edge search number of cographs, thereby resolving the computational complexity of edge searching on this graph class. To achieve this we give a characterization of the edge search number of the join of two graphs. With our result, the knowledge on graph searching of cographs is now complete: node, mixed, and edge search numbers of cographs can all be computed efficiently. Furthermore, we are one step closer to computing the edge search number of permutation graphs.

Cutwidth of Split Graphs, Threshold Graphs, and Proper Interval Graphs

We give a linear-time algorithm to compute the cutwidth of threshold graphs, thereby resolving the computational complexity of cutwidth on this graph class. Although our algorithm is simple and intuitive, its correctness proof relies on a series of non-trivial structural results, and turns out to be surprisingly complex. Threshold graphs are a well-studied subclass of interval graphs and of split graphs, both of which are unrelated subclasses of chordal graphs. To complement our result, we show that cutwidth is NP-complete on split graphs, and consequently also on chordal graphs. In addition, we show that cutwidth is trivial on proper interval graphs, another subclass of interval graphs. The cutwidth of interval graphs is open, and only very few graph classes are known so far on which polynomial-time cutwidth algorithms exist. Thus we contribute to define the border between graph classes on which cutwidth is polynomially solvable and on which it remains NP-complete.

A Self-stabilizing Algorithm for Graph Searching in Trees

Graph searching games have been extensively studied in the past years. The graph searching problem involves a team of searchers who are attempting to capture a fugitive moving along the edges of the graph. In this paper we consider the graph searching problem in a network environment, namely a tree network. Searchers are software programs and the fugitive is a virus that spreads rapidly. Every node of the network which the virus may have reached, becomes contaminated. The purpose of the game is to clean the network. In real world distributed systems faults can occur and thus it is desirable for an algorithm to be able to facilitate the cleaning of a network in an optimal way, and also to reconfigure on the fly. In this paper we give the first self-stabilizing algorithm for solving the graph searching problem in trees. Our algorithm stabilizes after O (n 3) time steps under the distributed adversarial daemon. Our algorithm solves the node searching variant of the graph searching problem, but can with small modifications also solve edge and mixed searching.

Comparing First-Fit and Next-Fit for online edge coloring

We study the performance of the algorithms First -Fit and Next -Fit for two online edge coloring problems. In the min-coloring problem, all edges must be colored using as few colors as possible. In the max-coloring problem, a fixed number of colors is given, and as many edges as possible should be colored. Previous analysis using the competitive ratio has not separated the performance of First -Fit and Next -Fit, but intuition suggests that First -Fit should be better than Next -Fit. We compare First -Fit and Next -Fit using the relative worst-order ratio, and show that First -Fit is better than Next -Fit for the min-coloring problem. For the max-coloring problem, we show that First -Fit and Next -Fit are not strictly comparable, i.e., there are graphs for which First -Fit is significantly better than Next -Fit and graphs where Next -Fit is slightly better than First -Fit.

Comparing First-Fit and Next-Fit for Online Edge Coloring

We study the performance of the algorithms First-Fit and Next-Fit for two online edge coloring problems. In the min-coloring problem, all edges must be colored using as few colors as possible. In the max-coloring problem, a fixed number of colors is given, and as many edges as possible should be colored. Previous analysis using the competitive ratio has not separated the performance of First-Fit and Next-Fit, but intuition suggests that First-Fit should be better than Next-Fit. We compare First-Fit and Next-Fit using the relative worst order ratio, and show that First-Fit is better than Next-Fit for the min-coloring problem. For the max-coloring problem, we show that First-Fit and Next-Fit are not strictly comparable, i.e., there are graphs for which First-Fit is better than Next-Fit and graphs where Next-Fit is slightly better than First-Fit.