Matjaž Kovše

Cover-Incomparability Graphs of Posets

Cover-incomparability graphs (C-I graphs, for short) are introduced, whose edge-set is the union of edge-sets of the incomparability and the cover graph of a poset. Posets whose C-I graphs are chordal (resp. distance-hereditary, Ptolemaic) are characterized in terms of forbidden isometric subposets, and a general approach for studying C-I graphs is proposed. Several open problems are also stated.

New results on variants of covering codes in Sierpiński graphs

In this paper we study identifying codes, locating-dominating codes, and total-dominating codes in Sierpiński graphs. We compute the minimum size of such codes in Sierpiński graphs.

Computing median and antimedian sets in median graphs

The median (antimedian) set of a profile π=(u1,…,uk) of vertices of a graph G is the set of vertices x that minimize (maximize) the remoteness ∑id(x,ui). Two algorithms for median graphs G of complexity O(n idim(G)) are designed, where n is the order and idim(G) the isometric dimension of G. The first algorithm computes median sets of profiles and will be in practice often faster than the other algorithm which in addition computes antimedian sets and remoteness functions and works in all partial cubes.

Lattice embeddings of trees

This note presents a linear algorithm that isometrically embeds a given tree T into an integer lattice of minimal dimension and allows one to compute the lattice coordinates of every single vertex of T in optimal time.

On Graph Identification Problems and the Special Case of Identifying Vertices Using Paths

In this paper, we introduce the identifying path cover problem: an identifying path cover of a graph G is a set \(\mathcal P\) of paths such that each vertex belongs to a path of \(\mathcal P\), and for each pair u,v of vertices, there is a path of \(\mathcal P\) which includes exactly one of u,v. This problem is related to a large variety of identification problems. We investigate the identifying path cover problem in some families of graphs. In particular, we derive the optimal size of an identifying path cover for paths, cycles, hypercubes and topologically irreducible trees and give an upper bound for all trees. We give lower and upper bounds on the minimum size of an identifying path cover for general graphs, and discuss their tightness. In particular, we show that any connected graph G has an identifying path cover of size at most \(\left\lceil\frac{2(|V(G)|-1)}{3}\right\rceil\). We also study the computational complexity of the associated optimization problem, in particular we show that when the length of the paths is asked to be of a fixed value, the problem is APX-complete.