No abstract

A temporal graph is a graph in which the edge set can change from step to step. The temporal graph exploration problem TEXP is the problem of computing a foremost exploration schedule for a temporal graph, i.e., a temporal walk that starts at a given start node, visits all nodes of the graph, and has the smallest arrival time. In the first part of the paper, we consider only undirected temporal graphs that are connected at each step. For such temporal graphs with n nodes, we show that it is NP-hard to approximate TEXP with ratio O(n 1−ε ) for any ε > 0. We also provide an explicit construction of temporal graphs that require Θ(n 2 ) steps to be explored. We then consider TEXP under the assumption that the underlying graph (i.e. the graph that contains all edges that are present in the temporal graph in at least one step) belongs to a specific class of graphs. Among other results, we show that temporal graphs can be explored in O(n 1.5 k 2 log n) steps if the underlying graph has treewidth k and in O(n log 3 n) steps if the underlying graph is a 2 × n grid. In the second part of the paper, we replace the connectedness assumption by a weaker assumption and show that m-edge temporal graphs with regularly present edges and with random edges can always be explored in O(m) steps and O(m log n) steps with high probability, respectively. We finally show that the latter result can be used to obtain a distributed algorithm for the gossiping problem.

We present space-efficient algorithms for computing cut vertices in a given graph with n vertices and m edges in linear time using O(n + min{m, n log log n}) bits. With the same time and using O(n+m) bits, we can compute the biconnected components of a graph. We use this result to show an algorithm for the recognition of (maximal) outerplanar graphs in O(n log log n) time using O(n) bits.

Many algorithms have been developed for NP-hard problems on graphs with small treewidth k. For example, all problems that are expressible in linear extended monadic second order can be solved in linear time on graphs of bounded treewidth. It turns out that the bottleneck of many algorithms for NP-hard problems is the computation of a tree decomposition of width O(k). In particular, by the bidimensional theory, there are many linear extended monadic second order problems that can be solved on n-vertex planar graphs with treewidth k in a time linear in n and subexponential in k if a tree decomposition of width O(k) can be found in such a time.We present the first algorithm that, on n-vertex planar graphs with treewidth k, finds a tree decomposition of width O(k) in such a time. In more detail, our algorithm has a running time of O(nk 2 log k). We show the result as a special case of a result concerning so-called weighted treewidth of weighted graphs.

Abstract. We introduce three new complexity parameters that in some sense measure how chordal-like a graph is. The similarity to chordal graphs is used to construct simple polynomial-time approximation algorithms with constant approximation ratio for many N P-hard problems, when restricted to graphs for which at least one of our new complexity parameters is bounded by a constant. As applications we present approximation algorithms with constant approximation ratio for maximum weighted independent set, minimum (independent) dominating set, minimum vertex coloring, maximum weighted clique, and minimum clique partition for large classes of intersection graphs.