Some remarks on the geodetic number of a graph

A set of vertices D of a graph G is geodetic if every vertex of G lies on a shortest path between two not necessarily distinct vertices in D. The geodetic number of G is the minimum cardinality of a geodetic set of G. We prove that it is NP-complete to decide for a given chordal or chordal bipartite graph G and a given integer k whether G has a geodetic set of cardinality at most k. Furthermore, we prove an upper bound on the geodetic number of graphs without short cycles and study the geodetic number of cographs, split graphs, and unit interval graphs.

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

[2]  Kellogg S. Booth,et al.  Dominating Sets in Chordal Graphs , 1982, SIAM J. Comput..

[3]  José Cáceres,et al.  On geodetic sets formed by boundary vertices , 2006, Discret. Math..

[4]  Peter L. Hammer,et al.  Difference graphs , 1990, Discret. Appl. Math..

[5]  Douglas B. West,et al.  A short proof that 'proper = unit' , 1998, Discret. Math..

[6]  Mustafa Atici,et al.  Computational Complexity of Geodetic Set , 2002, Int. J. Comput. Math..

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

[8]  Jeremy P. Spinrad,et al.  Ordered Vertex Partitioning , 2000, Discret. Math. Theor. Comput. Sci..

[9]  Frank Harary,et al.  Extremal problems in geodetic graph theory , 1998 .

[10]  Andreas Brandstädt,et al.  The NP-Completeness of Steiner Tree and Dominating Set for Chordal Bipartite Graphs , 1987, Theor. Comput. Sci..

[11]  Jeremy P. Spinrad,et al.  Linear-time modular decomposition and efficient transitive orientation of comparability graphs , 1994, SODA '94.

[12]  Gary Chartrand,et al.  The geodetic number of a graph: A survey , 2002 .

[13]  Tao Jiang,et al.  On the Steiner, geodetic and hull numbers of graphs , 2005, Discret. Math..

[14]  Byung Kee Kim,et al.  THE GEODETIC NUMBER OF A GRAPH , 2004 .

[15]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[16]  N. Sloane,et al.  Proof Techniques in Graph Theory , 1970 .