Triangle edge deletion on planar glasses-free RGB-digraphs

Abstract

In this paper, we revisit the triangle edge deletion problem a variant of feedback arc set problem which is one of the Karp's 21 NP-complete problems. Given a graph G(V,E), it is to find the edge set F such that |F|≤k and G′(V,E∖F) is a triangle-free graph. Yannakakis [1] showed it is hard for simple digraphs as well. The latest results showed that triangle edge deletion problem is still hard even for (a) planar undirected graphs of maximum degree seven [2] and (b) RGB-digraphs [3] which is a special class of digraphs. To make a further step, we show a new result in this paper that triangle edge deletion problem is still NP-complete even for planar RGB-digraphs of maximum degree eight with no glasses. On the other hand, we can find a kernel consisting of 11k/3 vertices for planar RGB-digraphs with no glasses. The results have an interesting application for the derivation of a tighter dichotomy on resilience decision problem in database theory.