Planar graph bipartization in linear time

Planar graph bipartization in linear time

For each constant k, we present a linear time algorithm that, given a planar graph G, either finds a minimum odd cycle vertex transversal in G or guarantees that there is no transversal of size at most k.

Computing the Girth of a Planar Graph in Linear Time

The girth of a graph is the minimum weight of all simple cycles of the graph. We study the problem of determining the girth of an n-node unweighted undirected planar graph. The first nontrivial algorithm for the problem, given by Djidjev, runs in O(n5/4 logn) time. Chalermsook, Fakcharoenphol, and Nanongkai reduced the running time to O(n log n). Weimann and Yuster further reduced the running t...

Algorithm Engineering for Optimal Graph Bipartization

We examine exact algorithms for the NP-hard Graph Bipartization problem. The task is, given a graph, to find a minimum set of vertices to delete to make it bipartite. Based on the “iterative compression” method introduced by Reed, Smith, and Vetta in 2004, we present new algorithms and experimental results. The worst-case time complexity is improved. Based on new structural insights, we give a ...

Level Planar Embedding in Linear Time

A level graph G = (V, E, φ) is a directed acyclic graph with a mapping φ : V → {1, 2, . . . , k}, k ≥ 1, that partitions the vertex set V as V = V 1∪V 2∪· · ·∪V , V j = φ−1(j), V i∩V j = ∅ for i 6= j, such that φ(v) ≥ φ(u) + 1 for each edge (u, v) ∈ E. The level planarity testing problem is to decide if G can be drawn in the plane such that for each level V , all v ∈ V i are drawn on the line l...

Recognizing Leveled-Planar Dags in Linear Time