A connected path decomposition of a simple graph G is a path decomposition (X1, . . . , X l ) such that the subgraph of G induced by X1 ∪ · · · ∪ Xi is connected for each i ∈ {1, . . . , l}. The connected pathwidth of G is then the minimum width over all connected path decompositions of G. We prove that for each fixed k, the connected pathwidth of any input graph can be computed in polynomial-time. This answers an open question raised by Fedor V. Fomin during the GRASTA 2017 workshop, since connected pathwidth is equivalent to the connected (monotone) node search game.

In this paper we study relations between connected and weakly convex domination numbers. We show that in general the difference between these numbers can be arbitrarily large and we focus on the graphs for which a weakly convex domination number equals a connected domination number. We also study the influence of the edge removing on the weakly convex domination number, in particular we prove that the weakly convex domination number is an interpolating function.

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This work considers the following extension of the tree-depth problem: for a given input graph G and integers k and b, find a rooted forest F of height at most k and width at most b (defined as the maximum number of vertices allowed in a level of F ) such that G is a subgraph of the closure of F . We are interested in the case when G is a line graph of a tree, proving that the problem is N P-hard and obtaining a polynomial-time additive 2b-approximation algorithm. This particular class of graphs received a significant attention in the past, mainly due to a number of potential applications it provides. These include applications in parallel processing, e.g., parallel assembly of modular products, or parallel query processing in relational databases, as well as purely combinatorial applications, including searching in tree-like partial orders (which in turn generalizes binary search on sorted data). The latter can be used for automated program testing.

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