Monotony Properties of Connected Visible Graph Searching

Search games are attractive for their correspondence with classical width parameters. For instance, the invisible search number (a.k.a. node search number) of a graph is equal to its pathwidth plus 1, and the visible search number of a graph is equal to its treewidth plus 1. The connected variants of these games ask for search strategies that are connected, i.e., at every step of the strategy, the searched part of the graph induces a connected subgraph. We focus on monotone search strategies, i.e., strategies for which every node is searched exactly once. It is known that the monotone connected visible search number of an n-node graph is at most O(log n) times its visible search number. First, we prove that this logarithmic bound is tight. Precisely, we prove that there is an infinite family of graphs for which the ratio monotone connected visible search number over visible search number is Ω(log n). Second, we prove that, as opposed to the non-connected variant of visible graph searching, ”recontamination helps” for connected visible search. Precisely, we describe an infinite family of graphs for which any monotone connected visible search strategy for any graph in this family requires strictly more searchers than the connected visible search number of the graph

DAG-width and parity games

Tree-width is a well-known metric on undirected graphs that measures how tree-like a graph is and gives a notion of graph decomposition that proves useful in algorithm development. Tree-width is characterised by a game known as the cops-and-robber game where a number of cops chase a robber on the graph. We consider the natural adaptation of this game to directed graphs and show that monotone strategies in the game yield a measure with an associated notion of graph decomposition that can be seen to describe how close a directed graph is to a directed acyclic graph (DAG). This promises to be useful in developing algorithms on directed graphs. In particular, we show that the problem of determining the winner of a parity game is solvable in polynomial time on graphs of bounded DAG-width. We also consider the relationship between DAG-width and other measures of such as entanglement and directed tree-width. One consequence we obtain is that certain NP-complete problems such as Hamiltonicity and disjoint paths are polynomial-time computable on graphs of bounded DAG-width

LIFO−Search on Digraphs: A Searching Game for Cycle−Rank

We consider the extension of the last-in-first-out graph searching game of Giannopoulou and Thilikos to digraphs. We show that all common variations of the game require the same number of searchers, and the minimal number of searchers required is one more than the cycle-rank of the digraph. We also obtain a tight duality theorem, giving a precise min-max characterization of obstructions for cycle-rank.Comment: 14 page

On Exact Algorithms for Treewidth

We give experimental and theoretical results on the problem of computing the treewidth of a graph by exact exponential time algorithms using exponential space or using only polynomial space. We first report on an implementation of a dynamic programming algorithm for computing the treewidth of a graph with running time ). This algorithm is based on the old dynamic programming method introduced by Held and Karp for the Traveling Salesman problem. We use some optimizations that do not a#ect the worst case running time but improve on the running time on actual instances and can be seen to be practical for small instances. However, our experiments show that the space used by the algorithm is an important factor to what input sizes the algorithm is e#ective. For this purpose, we settle the problem of computing treewidth under the restriction that the space used is only polynomial. In thi

Prädiktive Schwingungskompensation für Werkzeugmaschinen mit Parallelkinematik

International audienceWe introduce nondeterministic graph searching with a controlled amount of nondeterminism and show how this new tool can be used in algorithm design and combinatorial analysis applying to both pathwidth and treewidth. We prove equivalence between this game-theoretic approach and graph decompositions called q -branched tree decompositions, which can be interpreted as a parameterized version of tree decompositions. Path decomposition and (standard) tree decomposition are two extreme cases of q-branched tree decompositions. The equivalence between nondeterministic graph searching and q-branched tree decomposition enables us to design an exact (exponential time) algorithm computing q-branched treewidth for all q, which is thus valid for both treewidth and pathwidth. This algorithm performs as fast as the best known exact algorithm for pathwidth. Conversely, this equivalence also enables us to design a lower bound on the amount of nondeterminism required to search a graph with the minimum number of searchers

REPORTS IN INFORMATICS

ISSN 0333-3590 Graph searching, elimination trees, and ageneralization of bandwidt