Discussiones Mathematicae Graph Theory 33(4) (2013) 731-745
doi: 10.7151/dmgt.1700

[BIBTex] [PDF] [PS]

Path-neighborhood Graphs

R.C. Laskar

Department of Mathematical Sciences, Clemson University
Clemson, SC 29634, USA

Henry Martyn Mulder

Econometrisch Instituut, Erasmus Universiteit
P.O. Box 1738, 3000 DR Rotterdam, The Netherlands


A path-neighborhood graph is a connected graph in which every neighborhood induces a path. In the main results the 3-sun-free path-neighborhood graphs are characterized. The 3-sun is obtained from a 6-cycle by adding three chords between the three pairs of vertices at distance 2. A Pk-graph is a path-neighborhood graph in which every neighborhood is a Pk, where Pk is the path on k vertices. The Pk-graphs are characterized for k ≤ 4.

Keywords: path-neighborhood graph, outerplanar graph, MOP, snake, 3-sun, k-fun

2010 Mathematics Subject Classification: 05C75, 05C38, 05C99, 05C10.


[1]R. Balakrishnan, Triangle graphs, in: Graph Connections (Cochin, 1998), p. 44, Allied Publ., New Delhi, 1999.
[2]R. Balakrishnan, J. Bagga, R. Sampathkumar and N. Thillaigovindan, Triangle graphs, preprint.
[3]H.J. Bandelt and H.M. Mulder, Pseudo-median graphs: decompostion via amalgamation and Cartesian multiplication, Discrete Math. 94 (1991) 161--180, doi: 10.1016/0012-365X(91)90022-T.
[4]M. Borowiecki, P. Borowiecki, E. Sidorowicz and Z. Skupień, On extremal sizes of locally k-tree graphs, Czechoslovak Math. J.  60 (2010) 571--587, doi: 10.1007/s10587-010-0037-z.
[5]A. Brandstädt, V.B. Le, and J.P. Spinrad, Graph classes a survey (in: SIAM Monographs on Discrete Mathematics and Applications, Philadelphia, 1999).
[6]M. Brown and R. Connelly, On graphs with constant link, in: New directions in the theory of graphs, Proc. Third Ann Arbor Conf., Univ. Michigan, Ann Arbor, Mich., 1971, F. Harary Ed. (Academic Press, New York, 1973) 19--51, doi: 10.1016/0012-365X(75)90037-0.
[7]M. Brown and R. Connelly, On graphs with constant link II, Discrete Math. 11 (1975) 199--232.
[8]M. Brown and R. Connelly, Extremal problems for local properties of graphs, Australas. J. Combin. 4 (1991) 25--31.
[9]B.L. Chilton, R. Gould and A.D. Polimeni, A note on graphs whose neighborhoods are n-cycles, Geom. Dedicata 3 (1974) 289--294, doi: 10.1007/BF00181321.
[10]A.A. Di, doi: 10.1007/3-540-60692-0_40.
[11]Y. Egawa and R.E. Ramos, Triangle graphs, Math. Japon. 36 (1991) 465--467.
[12]D. Fronček, Locally linear graphs, Math. Slovaca 39 (1989) 3--6.
[13]D. Fronček, On graphs with prescribed edge neighbourhoods, Comment. Math. Univ. Carolin. 30 (1989) 749--754.
[14]D. Fronček, Locally path-like graphs, Časopis Pěst. Mat. 114 (1989) 176--180.
[15]M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Ann. Discrete Math. 57 (Elsevier, Amsterdam, 2004).
[16]J.I. Hall, Graphs with constnt link and small degree and order, J. Graph Theory 9 (1985) 419--444, doi: 10.1002/jgt.3190090313.
[17]F. Harary, Graph Theory (Addison-Wesley, Reading Massachusetts, 1969).
[18]P. Hell, Graphs with given neighborhoods I, in: Problémes combinatoires et théorie des graphes, Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976, (Colloq. Internat. CNRS, 260, CNRS, Paris, 1978) 219--223.
[19]R.E. Jamison, F.R. McMorris and H.M. Mulder, Graphs with only caterpillars as spanning trees, Discrete Math. 272 (2003) 81--95, doi: 10.1016/S0012-365X(03)00186-9.
[20]K. Kuratowski, Sur le probléme des courbes gauches en topologie, Fund. Math. 15 (1930) 271-?283.
[21]R.C. Laskar, H.M. Mulder and B. Novick, Maximal outerplanar graphs as chordal graphs, as path-neighborhood graphs, and as triangle graphs, Australas. J. Combin. 52 (2012) 185--195.
[22]R.C. Laskar and H.M. Mulder, Triangle graphs, EI-Report EI 2009-42, Econometrisch Instituut, Erasmus Universiteit, Rotterdam.
[23]C. Lekkerkerker and J. Boland, Representation of a graph by a finite set of intervals on the real line, Fund. Math. 51 (1962) 45--64.
[24]A. Márquez, A. de Mier, M. Noy and M.P. Revuelta, Locally grid graphs: classification and Tutte uniqueness, Discrete Math. 266 (2003) 327--352, doi: 10.1016/S0012-365X(02)00818-X.
[25]S.L. Mitchell, Algorithms on trees and maximal outerplanar graphs: design, complexity analysis and data structures study, PhD Thesis, University of Virginia, 1977.
[26]T.D. Parsons, Circulant graphs embeddings, J. Combin. Theory (B) 29 (1980) 310--320, doi: 10.1016/0095-8956(80)90088-X.
[27]T.D. Parsons and T. Pisanski, Graphs which are locally paths, in: Combinatorics and Graph Theory, Banach Center Publ., 25, (PWN, Warsaw, 1989) 127--135.
[28]N. Pullman, Clique covering of graphs IV. Algorithms, SIAM J. Comput. 13 (1984) 57--75, doi: 10.1137/0213005.
[29]S.M. Ulam, A collection of mathematical problems, in: Interscience Tracts in Pure and Applied Mathematics, Vol. XIII, No. 8, (Interscience Publishers, New York/London, 1960).
[30]B. Zelinka, Edge neighbourhood graphs. Czechoslovak Math. J.  36 (1986) 44--47.
[31]B. Zelinka, Locally snake-like graphs, Math. Slovaka 38 (1988) 85--88.
[32]A.A. Zykov, Graph-theoretical results of Novosibirsk mathematicians in: M. Fiedler ed., Theory of Graphs and its Applications, Proc. Sympos. Smolenice, 1963 (Publ. House Czechoslovak Acad. Sci., Prague 1964) 151--153.

Received 28 November 2011
Revised 29 August 2012
Accepted 10 September 2012