Perfect Graphs, Partitionable Graphs and Cutsets

Journal of Combinatorial Theory, Series B, 1985

We first establish a certain property of minimal imperfect graphs generate large classes of perfect graphs. 0 1985 Academic Press, Inc. and then use it 1. THE MAIN RESULT Claude Berge proposed to call a graph perfect if, for each of its induced subgraphs F, the chromatic number of F equals the largest number m(F) of pairwise adjacent vertices in F. A clique in a graph is any set of pairwise adjacent vertices; a stable set is any set of pairwise nonadjacent vertices. By a star-cutset in a graph G, we shall mean a nonempty set C of vertices such that G -C is disconnected and such that some vertex in C is adjacent to all the remaining vertices in C. THE STAR-CUTSET LEMMA No minimal imperfect graph has a star-cutset. Proof: Trivially, if G is a minimal imperfect graph then every proper induced subgraph of G is m(G)-colourable. and cu(G -S) = m(G) for every stable set of S in G. (2) We shall prove that no graph G with properties ( ) and ( ) has a star-cutset. For this purpose, consider an arbitrary graph G such that G satisfies (1) and G has a star-cutset C; we only need prove that G does not satisfy (2). Since G -C is disconnected, the set of its vertices splits into nonempty disjoint parts VI, v/2 such that no vertex in V, is adjacent to a vertex in V,. Let Gi (i = 1,2) be the subgraph of G induced by Vi u C; by (1 ), there is a colouring fi of Gi by o(G) colours. Since C is a star-cutset, some vertex w 189

2019

We look at some graph problems related to covering, partition, and connectivity. First, we study the problems of covering and partitioning edges with bicliques, especially from the viewpoint of parameterized complexity. For the partition problem, we develop much more efficient algorithms than the ones previously known. In contrast, for the cover problem, our lower bounds show that the known algorithms are probably optimal. Next, we move on to graph coloring, which is probably the most extensively studied partition problem in graphs. Hadwiger's conjecture is a long-standing open problem related to vertex coloring. We prove the conjecture for a special class of graphs, namely squares of 2-trees, and show that square graphs are important in connection with Hadwiger's conjecture. Then, we study a coloring problem that has been emerging recently, called rainbow coloring. This problem lies in the intersection of coloring and connectivity. We study different variants of rainbow coloring and present bounds and complexity results on them. Finally, we move on to another parameter related to connectivity called spanning tree congestion (STC). We give tight bounds for STC in general graphs and random graphs. While proving the results on STC, we also make some contributions to the related area of connected partitioning. Zusammenfassung Wir betrachten einige Graphprobleme mit Bezug auf Abdeckung, Partition und Konnektivität. Zunächst untersuchen wir Kantenabdeckung und-partition mit Bicliquen, insbesondere im Hinblick auf parametrisierte Komplexität. Für das Partitionierungsproblem entwickeln wir sehr viel effizientere Algorithmen als die bisher bekannten. Für das Abdeckungsproblem hingegen zeigen unsere unteren Schranken, dass die bereits bekannten Algorithmen wahrscheinlich optimal sind. Als Nächstes betrachten wir Graphfärbung, das wahrscheinlich am meisten untersuchte Partitionsproblem in Graphen. Hadwigers Vermutung ist ein seit Langem bestehendes offenes Problem bezüglich der Knotenfärbung. Wir beweisen diese Vermutung für eine spezielle Graphklasse, nämlich Quadrate von 2-Bäumen, und zeigen, dass Quadratgraphen wichtig in Bezug auf Hadwigers Vermutung sind. Danach untersuchen wir das Problem der sogenannten Regenbogenfärbung. Dieses erst kürzlich entstandene Problem liegt im Schnittpunkt der Probleme Färbung und Konnektivität. Wir untersuchen verschiedene Varianten der Regenbogenfärbung und zeigen Schranken und Komplexitätsergebnisse. Schließlich gehen wirüber zu einem weiteren Konnektivitätsparameter, der sogenannten spanning tree congestion (STC). Wir präsentieren scharfe Schranken für STC in allgemeinen und zufällig generierten Graphen. Unsere Ergebnisse in diesem Bereich leisten darüberhinaus einen Beitrag zu dem verwandten Gebiet der verbundenen Partitionierung. vi First of all I would like to thank my advisor Andreas Karrenbauer. I thank him for accepting me as a PhD student and guiding me through my PhD for the past 4 years. He has always given me the freedom to follow my own research interests, at the same time making sure that I don't go much out of the track. I thank him for his contributions, technical and otherwise to my thesis. I was very lucky to have Prof. Sunil Chandran visiting MPI-Informatics during my PhD term for about 18 months. He has been a co-author in most of my publications and has had a big effect on my shaping my research outlook. It has been a great pleasure to collaborate with him. I thank Erik Jan for all the technical input and for the encouragement he has given me. I thank Prof. Ragesh Jaiswal, who was my Masters supervisor for being a mentor at the start of my research career. I thank all my other co-authors Anita, Marco, Juho, Paloma, Pinar, and Sanming for their contributions. I thank Kurt Mehlhorn, for giving me the opportunity to be a PhD student at the Algorithms department, and also for building such an encouraging atmosphere in the group. I thank Pavel, for being the most friendly and helpful officemate for 4 years. I think I have learned a lot from Pavel. I also thank the other current and past members of D1:

2011

Some problems related to security in communication networks lead to consider a new type of connectivity in graphs, namely the confldential connectivity. In this paper we present a characterization of unbreakable graphs using the notion of weak decomposition and we give some applications of minimal unbreakable graphs. In fact, we showed that a graph G is confldentially connected if and only if it does not have a star cutset. We also showed that a minimal imperfect graph does not have a star cutset. We gave a constructive proof of the fact that every (fi;!)-partitionable graph is confldentially connected, for a superclass of minimal imperfect graphs.

Discrete Applied Mathematics, 2013

Let G be a graph on n vertices. We call a subset A of the vertex set

2005

Let F be a family of graphs. Two graphs G 1 = (V 1 , E 1), G 2 = (V 2 , E 2) are said to have the same F-structure if there is a bijection f : V 1 → V 2 such that a subset S induces a graph belonging to F in G 1 if and only if its image f (S) induces a graph belonging to F in G 2. We characterize those graphs which have the same {P 3 , P 3 }-structure, or the same {K 3 , K 3 }-structure. This characterization shows that graph H is perfect if and only if it has the {P 3 , P 3 }-structure of some perfect graph G. In proving the main result, we need and prove the following result, which is of independent interest: If a graph J is claw-free and co-claw-free, then either (i) J has at most nine vertices, or (ii) every component of J is a path or a hole, or (iii) every component of J is a path or a hole.