arXiv subjects

Sebastian S. Johann

Publications and source records attributed to Sebastian S. Johann.

2 recordsLinked to original sources

arXiv2020

Simultaneously Dominating all Spanning Trees of a Graph

We investigate the problem of simultaneously dominating all spanning trees of a given graph. We prove that on 2-connected graphs, a subset of the vertices dominates all spanning trees of the graph if and only if it is a vertex cover. Using this fact we present an exact algorithm that finds a simultaneous dominating set of minimum size using an oracle for finding a minimum vertex cover. The algorithm can be implemented to run in polynomial time on several graph classes, such as bipartite or chordal graphs. We prove that there is no polynomial time algorithm that finds a minimum simultaneous dominating set on perfect graphs, unless P=NP. Finally, we provide a 2-approximation algorithm for finding a minimum simultaneous dominating set.

math.CO↗

arXiv2020

On the Mixed Connectivity Conjecture of Beineke and Harary

The conjecture of Beineke and Harary states that for any two vertices which can be separated by $k$ vertices and $l$ edges for $l\geq 1$ but neither by $k$ vertices and $l-1$ edges nor $k-1$ vertices and $l$ edges there are $k+l$ edge-disjoint paths connecting these two vertices of which $k+1$ are internally disjoint. In this paper we consider this conjecture for $l=2$ and any $k\in \mathbb{N}$. Afterwards, we utilize this result to prove that the conjecture holds for all graphs of treewidth at most $3$ and all $k$ and $l$. We also show that it is NP-complete to decide whether two vertices can be separated by $k$ vertices and $l$ edges.

math.CO↗