Computing Well-Covered Vector Spaces of Graphs using Modular Decomposition

A graph is well-covered if all its maximal independent sets have the same cardinality. This well studied concept was introduced by Plummer in 1970 and naturally generalizes to the weighted case. Given a graph $G$, a real-valued vertex weight function $w$ is said to be a well-covered weighting of $G$ if all its maximal independent sets are of the same weight. The set of all well-covered weightings of a graph $G$ forms a vector space over the field of real numbers, called the well-covered vector space of $G$. Since the problem of recognizing well-covered graphs is $\mathsf{co}$-$\mathsf{NP}$-complete, the problem of computing the well-covered vector space of a given graph is $\mathsf{co}$-$\mathsf{NP}$-hard. Levit and Tankus showed in 2015 that the problem admits a polynomial-time algorithm in the class of claw-free graph. In this paper, we give two general reductions for the problem, one based on anti-neighborhoods and one based on modular decomposition, combined with Gaussian elimination. Building on these results, we develop a polynomial-time algorithm for computing the well-covered vector space of a given fork-free graph, generalizing the result of Levit and Tankus. Our approach implies that well-covered fork-free graphs can be recognized in polynomial time and also generalizes some known results on cographs.Comment: 25 page

Fair allocation of indivisible goods under conflict constraints

We consider the fair allocation of indivisible items to several agents and add a graph theoretical perspective to this classical problem. Thereby we introduce an incompatibility relation between pairs of items described in terms of a conflict graph. Every subset of items assigned to one agent has to form an independent set in this graph. Thus, the allocation of items to the agents corresponds to a partial coloring of the conflict graph. Every agent has its own profit valuation for every item. Aiming at a fair allocation, our goal is the maximization of the lowest total profit of items allocated to any one of the agents. The resulting optimization problem contains, as special cases, both {\sc Partition} and {\sc Independent Set}. In our contribution we derive complexity and algorithmic results depending on the properties of the given graph. We can show that the problem is strongly NP-hard for bipartite graphs and their line graphs, and solvable in pseudo-polynomial time for the classes of chordal graphs, cocomparability graphs, biconvex bipartite graphs, and graphs of bounded treewidth. Each of the pseudo-polynomial algorithms can also be turned into a fully polynomial approximation scheme (FPTAS).Comment: A preliminary version containing some of the results presented here appeared in the proceedings of IWOCA 2020. Version 3 contains an appendix with a remark about biconvex bipartite graph

Graphs where search methods are indistinguishable

The 7th Student Computer Science Research Conference is an answer to the fact that modern PhD and already Master level Computer Science programs foster early research activity among the students. The prime goal of the conference is to become a place for students to present their research work and hence further encourage students for an early research. Besides the conference also wants to establish an environment where students from different institutions meet, let know each other, exchange the ideas, and nonetheless make friends and research colleagues. At last but not least, the conference is also meant to be meeting place for students with senior researchers from institutions others than their own.Sedma Študentska konferenca na področju računalništva in informatike je odgovor na dejstvo, da so študenti 2. in 3. Bolonjske stopnje prisiljeni v raziskovalno delo že zelo zgodaj. Prvenstveni cilj te konference je nuditi možnost tem študentom, da predstavijo rezultate svojega raziskovalnega dela in jih vzpodbuditi za nadaljnje delo. Poleg tega želi konference vzpostaviti okolje, kjer se študenti iz različnih institucij srečujejo, spoznavajo, izmenjujejo ideje in na koncu koncev sklepajo prijateljstva, ki jim ostajajo v zrelih letih. Nenazadnje ta konferenca omogoča tudi, da se študenti srečujejo in navezujejo stike s starejšimi raziskovalci iz drugih institucij

Recognizing graph search trees

Graph searches and the corresponding search trees can exhibit important structural properties and are used in various graph algorithms. The problem of deciding whether a given spanning tree of a graph is a search tree of a particular search on this graph was introduced by Hagerup and Nowak in 1985, and independently by Korach and Ostfeld in 1989 where the authors showed that this problem is efficiently solvable for DFS trees. A linear time algorithm for BFS trees was obtained by Manber in 1990. In this paper we prove that the search tree problem is also in P for LDFS, in contrast to LBFS, MCS, and MNS, where we show NP-completeness. We complement our results by providing linear time algorithms for these searches on split graphs

On the end-vertex problem of graph searches

End vertices of graph searches can exhibit strong structural properties and are crucial for many graph algorithms. The problem of deciding whether a given vertex of a graph is an end-vertex of a particular search was first introduced by Corneil, K\"ohler and Lanlignel in 2010. There they showed that this problem is in fact NP-complete for LBFS on weakly chordal graphs. A similar result for BFS was obtained by Charbit, Habib and Mamcarz in 2014. Here, we prove that the end-vertex problem is NP-complete for MNS on weakly chordal graphs and for MCS on general graphs. Moreover, building on previous results, we show that this problem is linear for various searches on split and unit interval graphs