Tomáš Gavenčiak

Cop-win graphs with maximum capture-time

We present an upper bound n-4 for the maximum length of a cop and robber game (the capture-time) on a cop-win graph of order n. This bound matches the known lower bound. We analyze the structure of the class of all graphs attaining this maximum and describe an inductive construction of the entire class. A cop and robber game is a two-player vertex-to-vertex pursuit combinatorial game where the players stand on the vertices of a graph and alternate in moving to adjacent vertices. Cops goal is to capture the robber by occupying the same vertex as the robber, robbers goal is to avoid capture.

Firefighting on square, hexagonal, and triangular grids

In this paper, we consider the firefighter problem on a graph G = ( V , E ) that is either finite or infinite. Suppose that a fire breaks out at a given vertex v ? V . In each subsequent time unit, a firefighter protects one vertex which is not yet on fire, and then the fire spreads to all unprotected neighbors of the vertices on fire. The objective of the firefighter is to save as many vertices as possible (if G is finite) or to stop the fire from spreading (for an infinite case).The surviving rate ? ( G ) of a finite graph G is defined as the expected percentage of vertices that can be saved when a fire breaks out at a vertex of G that is selected uniformly random. For a finite square grid P n ? P n , we show that 5 / 8 + o ( 1 ) ? ? ( P n ? P n ) ? 67 243 / 105 300 + o ( 1 ) (leaving the gap smaller than 0.0136) and conjecture that the surviving rate is asymptotic to 5/8.We define the surviving rate for infinite graphs and prove it to be 1 / 4 for the infinite square grid, even for more than one (but finitely many) initial fires. For the infinite hexagonal grid we provide a winning strategy if two additional vertices can be protected at any point of the process, and we conjecture that the firefighter has no strategy to stop the fire without additional help. We also show how the speed of the spreading fire can be reduced by a constant multiplicative factor. For triangular grid, we show that two firefighters can slow down the fire in the same sense, which is relevant to the conjecture that two firefighters cannot contain the fire on the triangular grid, and also corrects a previous result of Fogarty (2003).

Fixed Parameter Complexity of Distance Constrained Labeling and Uniform Channel Assignment Problems

We study computational complexity of the class of distance-constrained graph labeling problems from the fixed parameter tractability point of view. The parameters studied are neighborhood diversity and clique width.

Cops and Robbers on Intersection Graphs

The game of cops and robber, introduced by Nowakowski and Winkler in 1983, is played by two players on a graph G, one controlling k cops and the other one robber, all positioned on V G . The players alternate in moving their pieces to distance at most 1 each. The cops win if they capture the robber, the robber wins by escaping indefinitely. The cop-number of G, that is the smallest k such that k cops win the game, has recently been a widely studied parameter.

Sorting by swaps with noisy comparisons

We study sorting of permutations by random swaps if the comparison operator is noisy. The noise is not associated with the underlying fitness but is inherent to the comparison operator. This type of fitness-independent noise has not been studied before in the community but is prototypical for comparison-based evolutionary algorithms, which often do not need to compute or approximate explicit fitness values. As quality measure, we compute the average fitness of the stationary distribution. To measure runtime, we compute the minimal number of steps after which the expected fitness approximates the average fitness of the stationary distribution. As mutations, we allow swaps of any two elements which have distance at most r. We give theoretical results for the extreme cases r = 1 and r = n, and experimental results for intermediate cases. We find a trade-off between faster convergence (for large r) and better average quality of the solution after convergence (for small r).

Cops and Robbers on String Graphs

The game of cops and robber, introduced by Nowakowski and Winkler in 1983, is played by two players on a graph. One controls k cops and the other a robber. The players alternate and move their pieces to the distance at most one. The cops win if they capture the robber, the robber wins by escaping indefinitely. The cop number of G is the smallest k such that k cops win the game.

Deciding first order properties of matroids

Frick and Grohe [J. ACM 48 (2006), 1184---1206] introduced a notion of graph classes with locally bounded tree-width and established that every first order property can be decided in almost linear time in such a graph class. Here, we introduce an analogous notion for matroids (locally bounded branch-width) and show the existence of a fixed parameter algorithm for first order properties in classes of regular matroids with locally bounded branch-width. To obtain this result, we show that the problem of deciding the existence of a circuit of length at most k containing two given elements is fixed parameter tractable for regular matroids.

Cops and Robbers on intersection graphs

Abstract The cop number of a graph G is the smallest k such that k cops win the game of cops and robber on G . We investigate the maximum cop number of geometric intersection graphs, which are graphs whose vertices are represented by geometric shapes and edges by their intersections. We establish the following dichotomy for previously studied classes of intersection graphs: • The intersection graphs of arc-connected sets in the plane (called string graphs) have cop number at most 15, and more generally, the intersection graphs of arc-connected subsets of a surface have cop number at most 10 g + 15 in case of orientable surface of genus g , and at most 10 g ′ + 15 in case of non-orientable surface of Euler genus g ′ . For more restricted classes of intersection graphs, we obtain better bounds: the maximum cop number of interval filament graphs is two, and the maximum cop number of outer-string graphs is between 3 and 4. • The intersection graphs of disconnected 2-dimensional sets or of 3-dimensional sets have unbounded cop number even in very restricted settings. For instance, it follows from known results that the cop number is unbounded on intersection graphs of two-element subsets of a line. We further show that it is also unbounded on intersection graphs of 3-dimensional unit balls, of 3-dimensional unit cubes or of 3-dimensional axis-aligned unit segments.

Parameterized complexity of distance labeling and uniform channel assignment problems

Abstract We rephrase the Distance labeling problem as a specific uniform variant of the Channel Assignment problem and show that the latter one is fixed parameter tractable when parameterized by the neighborhood diversity together with the largest weight. Consequently, the Distance labeling problem is FPT when parameterized by the neighborhood diversity, the maximum p i and k . This is indeed a more general answer to an open question of Fiala et al.: Parameterized complexity of coloring problems: Treewidth versus vertex cover. Finally, we show that the uniform variant of the Channel Assignment problem becomes NP -complete when generalized to graphs of bounded clique width.