On approximating the d-girth of a graph

For a finite, simple, undirected graph G and an integer d≥1, a mindeg-dsubgraph is a subgraph of G of minimum degree at least d. The d-girth of G, denoted by ^gd(G), is the minimum size of a mindeg-d subgraph of G. It is a natural generalization of the usual girth, which coincides with the 2-girth. The notion of d-girth was proposed by Erdos et al. (1988, 1990) [14,15] and Bollobás and Brightwell (1989) [8] over 25 years ago, and studied from a purely combinatorial point of view. Since then, no new insights have appeared in the literature. Recently, first algorithmic studies of the problem have been carried out by Amini et al. (2012a,b) [2,4]. The current article further explores the complexity of finding a small mindeg-d subgraph of a given graph (that is, approximating its d-girth), by providing new hardness results and the first approximation algorithms in general graphs, as well as analyzing the case where G is planar.