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 NP-hard and obtaining a polynomial-time additive b-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 .
The tree-depth problem can be seen as finding an elimination tree of minimum height for a given input graph G . We introduce a bicriteria generalization in which additionally the width of the elimination tree needs to be bounded by some input integer b . We are interested in the case when G is the line graph of a tree, proving that the problem is NP-hard and obtaining a polynomial-time additive 2b-approximation algorithm. This particular class of graphs received significant attention in the past, mainly due to a number of potential applications , e.g. in 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) .
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