Jakub Teska

FO model checking of interval graphs

We study the computational complexity of the FO model checking problem on interval graphs, i.e., intersection graphs of intervals on the real line. The main positive result is that this problem can be solved in time O(n logn) for n-vertex interval graphs with representations containing only intervals with lengths from a prescribed finite set. We complement this result by showing that the same is not true if the lengths are restricted to any set that is dense in some open subset, e.g., in the set (1, 1+e).

Closure, clique covering and degree conditions for Hamilton-connectedness in claw-free graphs

Abstract We strengthen the closure concept for Hamilton-connectedness in claw-free graphs, introduced by the second and fourth authors, such that the strong closure G M of a claw-free graph G is the line graph of a multigraph containing at most two triangles or at most one double edge. Using the concept of strong closure, we prove that a 3-connected claw-free graph G is Hamilton-connected if G satisfies one of the following: (i) G can be covered by at most 5 cliques, (ii) δ ( G ) ≥ 4 and G can be covered by at most 6 cliques, (iii) δ ( G ) ≥ 6 and G can be covered by at most 7 cliques. Finally, by reconsidering the relation between degree conditions and clique coverings in the case of the strong closure G M , we prove that every 3-connected claw-free graph G of minimum degree δ ( G ) ≥ 24 and minimum degree sum σ 8 ( G ) ≥ n + 50 (or, as a corollary, of order n ≥ 142 and minimum degree δ ( G ) ≥ n + 50 8 ) is Hamilton-connected. We also show that our results are asymptotically sharp.

FO Model Checking of Interval Graphs

We study the computational complexity of the FO model checking problem on interval graphs, i.e., intersection graphs of intervals on the real line. The main positive result is that FO model checking and successor-invariant FO model checking can be solved in time O(n log n) for n-vertex interval graphs with representations containing only intervals with lengths from a prescribed finite set. We complement this result by showing that the same is not true if the lengths are restricted to any set that is dense in an open subset, e.g., in the set (1, 1 + µ).

Divisibility conditions in almost Moore digraphs with selfrepeats

Abstract Moore digraph is a digraph with maximum out-degree d, diameter k and order M d , k = 1 + d + … + d k . Moore digraphs exist only in trivial cases if d = 1 (i.e., directed cycle C k ) or k = 1 (i.e., complete symmetric digraph). Almost Moore digraphs are digraphs of order one less than Moore bound. We shall present new properties of almost Moore digraphs with selfrepeats from which we prove nonexistence of almost Moore digraphs for some k and d.

On 2-walks in chordal planar graphs

A 2-walk is a closed spanning trail which uses every vertex at most twice. A graph is said to be chordal if each cycle different from a 3-cycle has a chord. We prove that every chordal planar graph G with toughness t(G)>34 has a 2-walk.

Bounding the distance among longest paths in a connected graph

It is easy to see that in a connected graph any 2 longest paths have a vertex in common. For k>=7, Skupien in [7] obtained a connected graph in which some k longest paths have no common vertex, but every k-1 longest paths have a common vertex. It is not known whether every 3 longest paths in a connected graph have a common vertex and similarly for 4, 5, and 6 longest path. In [5] the authors give an upper bound on distance among 3 longest paths in a connected graph. In this paper we give a similar upper bound on distance between 4 longest paths and also for k longest paths, in general.

Stability of Hereditary Graph Classes Under Closure Operations

If is a subclass of the class of claw-free graphs, then is said to be stable if, for any , the local completion of G at any vertex is also in . If is a closure operation that turns a claw-free graph into a line graph by a series of local completions and is stable, then for any . In this article, we study stability of hereditary classes of claw-free graphs defined in terms of a family of connected closed forbidden subgraphs. We characterize line graph preimages of graphs in families that yield stable classes, we identify minimal families that yield stable classes in the finite case, and we also give a general background for techniques for handling unstable classes by proving that their closure may be included into another (possibly stable) class.