Pure pairs. IV. Trees in bipartite graphs

Steiner trees in uniformly quasi-bipartite graphs

The area of approximation algorithms for the Steiner tree problem in graphs has seen continuous progress over the last years. Currently the best approximation algorithm has a performance ratio of 1.550. This is still far away from 1.0074, the largest known lower bound on the achievable performance ratio. As all instances resulting from known lower bound reductions are uniformly quasi-bipartite,...

Spanning $k$-trees of Bipartite Graphs

A tree is called a k-tree if its maximum degree is at most k. We prove the following theorem. Let k ⩾ 2 be an integer, and G be a connected bipartite graph with bipartition (A,B) such that |A| ⩽ |B| ⩽ (k − 1)|A| + 1. If σk(G) ⩾ |B|, then G has a spanning k-tree, where σk(G) denotes the minimum degree sum of k independent vertices of G. Moreover, the condition on σk(G) is sharp. It was shown by ...

Bipartite graphs with even spanning trees

Contracting chordal graphs and bipartite graphs to paths and trees

We study the following two graph modification problems: given a graph G and an integer k, decide whether G can be transformed into a tree or into a path, respectively, using at most k edge contractions. These problems, which we call Tree Contraction and Path Contraction, respectively, are known to be NP-complete in general. We show that on chordal graphs these problems can be solved in O(n + m)...

Even and odd pairs in linegraphs of bipartite graphs

Two vertices in a graph are called an even pair (odd pair) if all induced paths between these two vertices have even (odd) length. Even and odd pairs have turned out to be of importance in conjunction with perfect graphs. We will characterize all linegraphs of bipartite graphs that contain an even resp. odd pair. In general, it is a co-NP-complete problem to decide whether a graph contains an e...