Resolving sets for Johnson and Kneser graphs

A set of vertices S in a graph G is a resolving set for G if, for any two vertices u,v, there exists [email protected]?S such that the distances d(u,x) d(v,x). In this paper, we consider the Johnson graphs J(n,k) and Kneser graphs K(n,k), and obtain various constructions of resolving sets for these graphs. As well as general constructions, we show that various interesting combinatorial objects can be used to obtain resolving sets in these graphs, including (for Johnson graphs) projective planes and symmetric designs, as well as (for Kneser graphs) partial geometries, Hadamard matrices, Steiner systems and toroidal grids.

[1]  Kaishun Wang,et al.  On the metric dimension of bilinear forms graphs , 2011, Discret. Math..

[2]  M. L. Puertas,et al.  Hypergraphs for computing determining sets of Kneser graphs , 2011, 1111.3252.

[3]  Delia Garijo,et al.  The determining number of Kneser graphs , 2013, Discret. Math. Theor. Comput. Sci..

[4]  Kaishun Wang,et al.  Metric dimension of some distance-regular graphs , 2011, J. Comb. Optim..

[5]  Azriel Rosenfeld,et al.  Landmarks in Graphs , 1996, Discret. Appl. Math..

[6]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[7]  J. Thas,et al.  Finite Generalized Quadrangles , 2009 .

[8]  Ortrud R. Oellermann,et al.  The metric dimension of Cayley digraphs , 2006, Discret. Math..

[9]  N. Duncan Leaves on trees , 2014 .

[10]  Ping Zhang,et al.  The metric dimension of unicyclic graphs , 2002 .

[11]  Mario Valencia-Pabon,et al.  On the diameter of Kneser graphs , 2005, Discret. Math..

[12]  Ioan Tomescu,et al.  Metric bases in digital geometry , 1984, Comput. Vis. Graph. Image Process..

[13]  László Babai,et al.  On the Complexity of Canonical Labeling of Strongly Regular Graphs , 1980, SIAM J. Comput..

[14]  Navin M. Singhi,et al.  Projective planes I , 2010, Eur. J. Comb..

[15]  Vasek Chvátal,et al.  Mastermind , 1983, Comb..

[16]  Prince Camille de Polignac On a Problem in Combinations , 1866 .

[17]  Leonard M. Blumenthal,et al.  Theory and applications of distance geometry , 1954 .

[18]  E. Lander Symmetric Designs: An Algebraic Approach , 1983 .

[19]  A. Barlotti,et al.  Combinatorics of Finite Geometries , 1975 .

[20]  L. Babai On the Order of Uniprimitive Permutation Groups , 1981 .

[21]  Gary Chartrand,et al.  Resolvability in graphs and the metric dimension of a graph , 2000, Discret. Appl. Math..

[22]  Debra L. Boutin Identifying Graph Automorphisms Using Determining Sets , 2006, Electron. J. Comb..

[23]  C. Colbourn,et al.  Handbook of Combinatorial Designs , 2006 .

[24]  Delia Garijo,et al.  On determining number and metric dimension of graphs , 2007 .

[25]  P. Cameron,et al.  Base size, metric dimension and other invariants of groups and graphs , 2011 .

[26]  András Sebö,et al.  On Metric Generators of Graphs , 2004, Math. Oper. Res..

[27]  N. Biggs SOME ODD GRAPH THEORY , 1979 .

[28]  David R. Wood,et al.  On the Metric Dimension of Cartesian Products of Graphs , 2005, SIAM J. Discret. Math..

[29]  Patric R. J. Östergård,et al.  There are exactly five biplanes with k=11 , 2006, Electron. Notes Discret. Math..

[30]  H. Weyl Permutation Groups , 2022 .

[31]  Behruz Tayfeh-Rezaie,et al.  A Hadamard matrix of order 428 , 2005 .

[32]  A. Hora,et al.  Distance-Regular Graphs , 2007 .

[33]  Karen Meagher,et al.  On the metric dimension of Grassmann graphs , 2010, Discret. Math. Theor. Comput. Sci..