Marek Tesař

The planar slope number of planar partial 3-trees of bounded degree

It is known that every planar graph has a planar embedding where edges are represented by non-crossing straight-line segments. We study the planar slope number, i.e., the minimum number of distinct edge-slopes in such a drawing of a planar graph with maximum degree Δ. We show that the planar slope number of every series-parallel graph of maximum degree three is three. We also show that the planar slope number of every planar partial 3-tree and also every plane partial 3-tree is at most

Clustered Planarity: Clusters with Few Outgoing Edges

2^{{\mathcal O}(\Delta)}

Complexity of locally injective homomorphism to the theta graphs

. In particular, we answer the question of Dujmovic et al. [Computational Geometry 38 (3), pp. 194–212 (2007)] whether there is a function f such that plane maximal outerplanar graphs can be drawn using at most f(Δ) slopes.

Rainbow colouring of split graphs

We present a linear algorithm for c-planarity testing of clustered graphs, in which every cluster has at most four outgoing edges.

Locally constrained homomorphisms on graphs of bounded treewidth and bounded degree

A Theta graph is a multigraph which is a union of at least three internally disjoint paths that have the same two distinct end vertices. In this extended abstract we show full computational complexity characterization of the problem of deciding the existence of a locally injective homomorphism from an input graph G to any fixed Theta graph.

Locally injective homomorphism to the simple weight graphs

A rainbow path in an edge coloured graph is a path in which no two edges are coloured the same. A rainbow colouring of a connected graph G is a colouring of the edges of G such that every pair of vertices in G is connected by at least one rainbow path. The minimum number of colours required to rainbow colour G is called its rainbow connection number. It is known that, unless P = NP, the rainbow connection number of a graph cannot be approximated in polynomial time to a multiplicative factor less than 5/4, even when the input graph is chordal Chandran and Rajendraprasad, FSTTCS 2013]. In this article, we determine the computational complexity of the above problem on successively more restricted graph classes, viz.: split graphs and threshold graphs. In particular, we establish the following: 1. The problem of deciding whether a given split graph can be rainbow coloured using k colours is NP-complete for k is an element of {2, 3}, but can be solved in polynomial time for all other values of k. Furthermore, any split graph can be rainbow coloured in linear time using at most one more colour than the optimum. 2. For every positive integer k, threshold graphs with rainbow connection number k can be characterised based on their degree sequence alone. Furthermore, we can optimally rainbow colour a threshold graph in linear time

Computational complexity of covering three-vertex multigraphs

A homomorphism from a graph G to a graph H is locally bijective, surjective, or injective if its restriction to the neighborhood of every vertex of G is bijective, surjective, or injective, respectively. We prove that the problems of testing whether a given graph G allows a homomorphism to a given graph H that is locally bijective, surjective, or injective, respectively, are NP-complete, even when G has pathwidth at most 5, 4, or 2, respectively, or when both G and H have maximum degree 3. We complement these hardness results by showing that the three problems are polynomial-time solvable if G has bounded treewidth and in addition G or H has bounded maximum degree.

Computational Complexity of Covering Three-Vertex Multigraphs

A Weight graph is a connected (multi)graph with two vertices u and v of degree at least three and other vertices of degree two. Moreover, if any of these two vertices is removed, the remaining graph contains a cycle. A Weight graph is called simple if the degree of u and v is three. We show full computational complexity characterization of the problem of deciding the existence of a locally injective homomorphism from an input graph G to any fixed simple Weight graph by identifying some polynomial cases and some NP-complete cases.

Dichotomy of the H-Quasi-Cover Problem

A covering projection from a graph G onto a graph H is a mapping of the vertices of G onto the vertices of H such that, for every vertex v of G, the neighborhood of v is mapped bijectively onto the neighborhood of its image. Moreover, if G and H are multigraphs, then this local bijection has to preserve multiplicities of the neighbors as well. The notion of covering projection stems from topology, but has found applications in areas such as the theory of local computation and construction of highly symmetric graphs. It provides a restrictive variant of the constraint satisfaction problem with additional symmetry constraints on the behavior of the homomorphisms of the structures involved.We investigate the computational complexity of the problem of deciding the existence of a covering projection from an input graph G to a fixed target graph H. Among other partial results this problem has been shown NP-hard for simple regular graphs H of valency greater than 2, and a full characterization of computational complexity has been shown for target multigraphs with 2 vertices. We extend the previously known results to the ternary case, i.e., we give a full characterization of the computational complexity in the case of multigraphs with 3 vertices. We show that even in this case a P/NP-completeness dichotomy holds.

The Planar Slope Number of Planar Partial 3-Trees of Bounded Degree

A covering projection from a graph G to a graph H is a mapping of the vertices of G to the vertices of H such that, for every vertex v of G, the neighborhood of v is mapped bijectively to the neighborhood of its image. Moreover, if G and H are multigraphs, then this local bijection has to preserve multiplicities of the neighbors as well. The notion of covering projection stems from topology, but has found applications in areas such as the theory of local computation and construction of highly symmetric graphs. It provides a restrictive variant of the constraint satisfaction problem with additional symmetry constraints on the behavior of the homomorphisms of the structures involved.