Laminar Structure of Ptolemaic Graphs and Its Applications

Ptolemaic graphs are graphs that satisfy the Ptolemaic inequality for any four vertices. The graph class coincides with the intersection of chordal graphs and distance hereditary graphs, and it is a natural generalization of block graphs (and hence trees). In this paper, a new characterization of ptolemaic graphs is presented. It is a laminar structure of cliques, and leads us to a canonical tree representation, which gives a simple intersection model for ptolemaic graphs. The tree representation is constructed in linear time from a perfect elimination ordering obtained by the lexicographic breadth first search. Hence the recognition and the graph isomorphism for ptolemaic graphs can be solved in linear time. Using the tree representation, we also give an O(n) time algorithm for the Hamiltonian cycle problem.

[1]  Marina Moscarini,et al.  Distance-Hereditary Graphs, Steiner Trees, and Connected Domination , 1988, SIAM J. Comput..

[2]  Tsan-sheng Hsu,et al.  The Hamiltonian problem on distance-hereditary graphs , 2006, Discret. Appl. Math..

[3]  Catriel Beeri,et al.  On the Desirability of Acyclic Database Schemes , 1983, JACM.

[4]  Feodor F. Dragan,et al.  A linear-time algorithm for connected r-domination and Steiner tree on distance-hereditary graphs , 1998, Networks.

[5]  Peter L. Hammer,et al.  Completely separable graphs , 1990, Discret. Appl. Math..

[6]  Jeremy P. Spinrad,et al.  Efficient graph representations , 2003, Fields Institute monographs.

[7]  A. Brandstädt,et al.  Graph Classes: A Survey , 1987 .

[8]  Dieter Kratsch,et al.  Graph-Theoretic Concepts in Computer Science , 1987, Lecture Notes in Computer Science.

[9]  M. Golumbic.  Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57) , 2004 .

[10]  Marina Moscarini,et al.  On Hypergraph Acyclicity and Graph Chordality , 1988, Inf. Process. Lett..

[11]  E. Howorka.  A CHARACTERIZATION OF DISTANCE-HEREDITARY GRAPHS , 1977 .

[12]  Jens Vygen,et al.  The Book Review Column1 , 2020, SIGACT News.

[13]  Ton Kloks,et al.  A Linear Time Algorithm for Minimum Fill-in and Treewidth for Distance Hereditary Graphs , 2000, Discret. Appl. Math..

[14]  Robert E. Tarjan,et al.  Algorithmic Aspects of Vertex Elimination on Graphs , 1976, SIAM J. Comput..

[15]  Lorna Stewart,et al.  A Linear Recognition Algorithm for Cographs , 1985, SIAM J. Comput..

[16]  Falk Nicolai,et al.  Homogeneous sets and domination: A linear time algorithm for distance - hereditary graphs , 2001, Networks.

[17]  Maw-Shang Chang,et al.  Linear-time algorithms for the Hamiltonian problems on distance-hereditary graphs, , 2005, Theor. Comput. Sci..

[18]  Gen-Huey Chen,et al.  Dynamic Programming on Distance-Hereditary Graphs , 1997, ISAAC.

[19]  Robert E. Tarjan,et al.  Simple Linear-Time Algorithms to Test Chordality of Graphs, Test Acyclicity of Hypergraphs, and Selectively Reduce Acyclic Hypergraphs , 1984, SIAM J. Comput..

[20]  Edward Howorka.  A characterization of ptolemaic graphs , 1981, J. Graph Theory.

[21]  Hong-Gwa Yeh,et al.  Centers and medians of distance-hereditary graphs , 2003, Discret. Math..

[22]  Martin Farber,et al.  Independent domination in chordal graphs , 1982, Oper. Res. Lett..

[23]  Philip N. Klein.  Efficient Parallel Algorithms for Chordal Graphs , 1996, SIAM J. Comput..

[24]  Michel Habib,et al.  A simple paradigm for graph recognition: application to cographs and distance hereditary graphs , 2001, Theor. Comput. Sci..

[25]  Fanica Gavril,et al.  Algorithms for Minimum Coloring, Maximum Clique, Minimum Covering by Cliques, and Maximum Independent Set of a Chordal Graph , 1972, SIAM J. Comput..

[26]  M. R. Rao,et al.  Combinatorial Optimization , 1992, NATO ASI Series.

[27]  Ryuhei Uehara,et al.  Efficient Algorithms for the Longest Path Problem , 2004, ISAAC.

[28]  Hans-Jürgen Bandelt,et al.  Distance-hereditary graphs , 1986, J. Comb. Theory, Ser. B.

[29]  Derek G. Corneil,et al.  Lexicographic Breadth First Search - A Survey , 2004, WG.

[30]  Ronald Fagin,et al.  Degrees of acyclicity for hypergraphs and relational database schemes , 1983, JACM.

[31]  Hong-Gwa Yeh,et al.  Domination in distance-hereditary graphs , 2002, Discret. Appl. Math..

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

[33]  J. Köbler,et al.  The Graph Isomorphism Problem: Its Structural Complexity , 1993 .