A certifying and dynamic algorithm for the recognition of proper circular-arc graphs

We present a dynamic algorithm for the recognition of proper circular-arc (PCA) graphs, that supports the insertion and removal of vertices (together with its incident edges). The main feature of the algorithm is that it outputs a minimally non-PCA induced subgraph when the insertion of a vertex fails. Each operation cost $O(\log n + d)$ time, where $n$ is the number vertices and $d$ is the degree of the modified vertex. When removals are disallowed, each insertion is processed in $O(d)$ time. The algorithm also provides two constant-time operations to query if the dynamic graph is proper Helly (PHCA) or proper interval (PIG). When the dynamic graph is not PHCA (resp. PIG), a minimally non-PHCA (resp. non-PIG) induced subgraph is obtained.

[1]  Emeric Gioan,et al.  Split decomposition and graph-labelled trees: characterizations and fully-dynamic algorithms for totally decomposable graphs , 2008, Discret. Appl. Math..

[2]  Leonidas Palios,et al.  A fully dynamic algorithm for the recognition of P4-sparse graphs , 2006, Theor. Comput. Sci..

[3]  Kurt Mehlhorn,et al.  Certifying algorithms , 2011, Comput. Sci. Rev..

[4]  Jing Huang,et al.  On the Structure of Local Tournaments , 1995, J. Comb. Theory, Ser. B.

[5]  Xiaotie Deng,et al.  Linear-Time Representation Algorithms for Proper Circular-Arc Graphs and Proper Interval Graphs , 1996, SIAM J. Comput..

[6]  Roded Sharan,et al.  A Fully Dynamic Algorithm for Recognizing and Representing Proper Interval Graphs , 2001, SIAM J. Comput..

[7]  Louis Ibarra,et al.  A Fully Dynamic Graph Algorithm for Recognizing Proper Interval Graphs , 2009, WALCOM.

[8]  R. Möhring.  Algorithmic graph theory and perfect graphs , 1986 .

[9]  Derek G. Corneil,et al.  A simple 3-sweep LBFS algorithm for the recognition of unit interval graphs , 2004, Discret. Appl. Math..

[10]  Louis Ibarra,et al.  A Fully Dynamic Graph Algorithm for Recognizing Interval Graphs , 2010, Algorithmica.

[11]  Louis Ibarra,et al.  Fully dynamic algorithms for chordal graphs and split graphs , 2008, TALG.

[12]  Derek G. Corneil,et al.  An Optimal, Edges-Only Fully Dynamic Algorithm for Distance-Hereditary Graphs , 2007, STACS.

[13]  Alan Tucker,et al.  Structure theorems for some circular-arc graphs , 1974, Discret. Math..

[14]  Gregory F. Sullivan,et al.  On-line error monitoring for several data structures , 1995, Twenty-Fifth International Symposium on Fault-Tolerant Computing. Digest of Papers.

[15]  Christophe Paul,et al.  Fully dynamic recognition algorithm and certificate for directed cographs , 2006, Discret. Appl. Math..

[16]  Roded Sharan,et al.  A fully dynamic algorithm for modular decomposition and recognition of cographs , 2004, Discret. Appl. Math..

[17]  Francisco J. Soulignac.  Fully Dynamic Recognition of Proper Circular-Arc Graphs , 2013, Algorithmica.

[18]  Christophe Paul,et al.  Fully Dynamic Algorithm for Recognition and Modular Decomposition of Permutation Graphs , 2008, Algorithmica.

[19]  Christophe Crespelle.  Fully Dynamic Representations of Interval Graphs , 2009, WG.

[20]  Leonidas Palios,et al.  A Fully Dynamic Algorithm for the Recognition of P4-Sparse Graphs , 2006, WG.

[21]  Haim Kaplan,et al.  Certifying algorithms for recognizing proper circular-arc graphs and unit circular-arc graphs , 2009, Discret. Appl. Math..

[22]  Wen-Lian Hsu,et al.  Linear Time Algorithms on Circular-Arc Graphs , 1991, Inf. Process. Lett..

[23]  Pavol Hell,et al.  Certifying LexBFS Recognition Algorithms for Proper Interval Graphs and Proper Interval Bigraphs , 2005, SIAM J. Discret. Math..

[24]  Pinar Heggernes,et al.  Dynamically maintaining split graphs , 2009, Discret. Appl. Math..

[25]  Celina M. H. de Figueiredo,et al.  A Linear-Time Algorithm for Proper Interval Graph Recognition , 1995, Inf. Process. Lett..

[26]  C. Lekkeikerker,et al.  Representation of a finite graph by a set of intervals on the real line , 1962 .

[27]  Jayme Luiz Szwarcfiter,et al.  Characterizations and recognition of circular-arc graphs and subclasses: A survey , 2009, Discret. Math..

[28]  Kurt Mehlhorn,et al.  Certifying algorithms for recognizing interval graphs and permutation graphs , 2003, SODA '03.

[29]  Daniel Meister,et al.  Recognition and computation of minimal triangulations for AT-free claw-free and co-comparability graphs , 2005, Discret. Appl. Math..

[30]  Stephan Olariu,et al.  Simple Linear Time Recognition of Unit Interval Graphs , 1995, Inf. Process. Lett..