Seminar iz diskretne matematike

Vodji seminarja: Boštjan Brešar in Sandi Klavžar
Predavanja potekajo ob ponedeljkih ob 15:15 v seminarski sobi P1 (Gosposvetska cesta 84, v 4. nadstropju).

Bodoča predavanja

Minula predavanja

Distinguishing trees and subcubic graphs

Predavatelj: Wilfried Imrich

Abstract: Thomas Tucker’s infinite motion conjecture asserts that the vertices of every connected, locally finite graph G can be colored with 2 colors such that the identity automorphism is the only automorphism that respects the coloring under the condition that every automorphism moves infinitely many vertices. We say G is 2-distinguishable if it has infinite motion.

The conjecture is true for many classes of graphs, but it is not known whether it holds for graphs with given maximum degree k, unless k=3.
Such graphs are called subcubic. However, there is evidence that infinite motion is not needed for subcubic graphs and that one can always find a 2-coloring that fixes all vertices with the exception of at most two pairs of interchangeable vertices. We show that this is true for trees, subcubic graphs without small cycles and vertex transitive cubic graphs.

In particular, we show that every subcubic infinite tree is 2-distinguishable and that every finite subcubic tree has a 2-coloring, which fixes all vertices, with the possible exception of two vertices of degree 1 with a common neighbor. We also show that the only vertex or edge-transitive finite or infinite subcubic graphs are the K_4, the K_{3,3} or the cube.

For finite or infinite trees of maximum valence k we show that there is a 2-coloring that fixes all vertices that have no endvertex within distance log_2 k +1 .

Acknowledgement: This joint work with Thomas Lachmann and Gundelinde Wiegel (vertex transitive graphs), Svenja Hüning, Judith Kloas and Hannah Schreiber (trees) and Thomas Tucker.

Speaker: Wilfried Imrich and possibly one of the co-authors.

Vector connectivity in graphs

Predavatelj: Martin Milanič

Motivated by challenges related to domination, connectivity, and information propagation in social and other networks, we study the Vector Connectivity problem. This problem, introduced by Boros et al. in 2013, takes as input a graph G and an integer r(v) for every vertex v of G, and the objective is to find a vertex subset S of minimum cardinality such that every vertex v either belongs to S, or is connected to at least r(v) vertices of S by disjoint paths. If we require each path to be of length exactly 1, we get the well-known vector domination problem, which is a generalization of the dominating set and vertex cover problems. Consequently, the vector connectivity problem becomes NP-hard if an upper bound on the length of the disjoint paths is also supplied as input. Due to the hardness of these domination variants even on restricted graph classes, like split graphs, Vector Connectivity seems to be a natural problem to study for drawing the boundaries of tractability for this type of problems.

In the talk, I will give an overview of known complexity results for the Vector Connectivity problem. In particular, the problem can be solved in polynomial time on split graphs, in addition to cographs and trees. On the other hand, the problem is NP-hard for planar line graphs and for planar bipartite graphs, APX-hard on general graphs, and can be approximated in polynomial time within a factor of ln(n) + 2 on all n-vertex graphs. Vertex covers and dominating sets in a graph G can be easily characterized as hitting sets of derived hypergraphs (of G itself, and of the closed neighborhood hypergraph of G, respectively). Using Menger’s Theorem, a similar characterization of vector connectivity sets can be derived.

Based on joint works with Endre Boros, Ferdinando Cicalese, Pinar Heggernes, Pim van ‘t Hof, and Romeo Rizzi.