On approximating the d-girth of a graph

For a finite, simple, undirected graph G and an integer d ≥ 1, a mindeg-d subgraph is a subgraph of G of minimum degree at least d. The d-girth of G, denoted 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. [13, 14] and Bollob10:03 AM 2/4/2011aacute;s and Brightwell [7] over 20 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 [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.

[1]  Michael R. Fellows,et al.  Parameterized Complexity , 1998 .

[2]  David S. Johnson,et al.  The Rectilinear Steiner Tree Problem is NP Complete , 1977, SIAM Journal of Applied Mathematics.

[3]  André E. Kézdy.  Studies in connectivity , 1991 .

[4]  R. Tarjan,et al.  A Separator Theorem for Planar Graphs , 1977 .

[5]  Bruce A. Reed,et al.  Degree constrained subgraphs , 2005, Discret. Appl. Math..

[6]  Shin-Ichi Nakano,et al.  Efficient Generation of Plane Triangulations without Repetitions , 2001, ICALP.

[7]  P. Erdos,et al.  Cycles in graphs without proper subgraphs of minimum degree 3 , 2022 .

[8]  Omid Amini,et al.  Parameterized complexity of finding small degree-constrained subgraphs , 2012, J. Discrete Algorithms.

[9]  David Peleg,et al.  On approximating the d-girth of a graph , 2013, Discret. Appl. Math..

[10]  Joseph Cheriyan,et al.  Approximating Minimum-Size k-Connected Spanning Subgraphs via Matching , 1998, Electron. Colloquium Comput. Complex..

[11]  David R. Karger,et al.  On approximating the longest path in a graph , 1997, Algorithmica.

[12]  Stéphane Pérennes,et al.  On the approximability of some degree-constrained subgraph problems , 2012, Discret. Appl. Math..

[13]  Stéphane Pérennes,et al.  Degree-Constrained Subgraph Problems: Hardness and Approximation Results , 2008, WAOA.

[14]  Jean-Claude Bermond,et al.  Induced Subgraphs of the Power of a Cycle , 1989, SIAM J. Discret. Math..

[15]  W. T. Tutte,et al.  A Census of Planar Triangulations , 1962, Canadian Journal of Mathematics.

[16]  Zeev Nutov,et al.  Approximating directed weighted-degree constrained networks , 2008, Theor. Comput. Sci..

[17]  Stéphane Pérennes,et al.  Hardness and Approximation of Traffic Grooming , 2007, ISAAC.

[18]  Ignasi Sau.  Optimization in Graphs under Degree Constraints. Application to Telecommunication Networks , 2009 .

[19]  Béla Bollobás,et al.  Long cycles in graphs with no subgraphs of minimal degree 3 , 1989, Discret. Math..

[20]  Richard H. Schelp,et al.  Subgraphs of minimal degree k , 1990, Discret. Math..

[21]  MohammadAli Safari,et al.  A Constant Factor Approximation for Minimum λ-Edge-Connected k-Subgraph with Metric Costs , 2011, SIAM J. Discret. Math..

[22]  Frederic Dorn,et al.  Planar Subgraph Isomorphism Revisited , 2009, STACS.

[23]  Harold N. Gabow,et al.  An efficient reduction technique for degree-constrained subgraph and bidirected network flow problems , 1983, STOC.

[24]  Brenda S. Baker,et al.  Approximation algorithms for NP-complete problems on planar graphs , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[25]  F. Clarke.  On _{_{*}()}(_{*}(), _{*}()) , 1979 .

[26]  Philip J. Lin,et al.  The ring grooming problem , 2004, Networks.

[27]  Ignasi Sau Valls,et al.  Optimization in graphs under degree constraints. application to telecommunication networks , 2009 .

[28]  Omid Amini,et al.  Parameterized Complexity of the Smallest Degree-Constrained Subgraph Problem , 2008, IWPEC.

[29]  Reinhard Diestel,et al.  Graph Theory , 1997 .