JOURNAL BROWSE
Search
Advanced SearchSearch Tips
GRAPHS WITH ONE HOLE AND COMPETITION NUMBER ONE
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
GRAPHS WITH ONE HOLE AND COMPETITION NUMBER ONE
KIM SUH-RYUNG;
  PDF(new window)
 Abstract
Let D be an acyclic digraph. The competition graph of D has the same set of vertices as D and an edge between vertices u and v if and only if there is a vertex x in D such that (u, x) and (v, x) are arcs of D. The competition number of a graph G, denoted by k(G), is the smallest number k such that G together with k isolated vertices is the competition graph of an acyclic digraph. It is known to be difficult to compute the competition number of a graph in general. Even characterizing the graphs with competition number one looks hard. In this paper, we continue the work done by Cho and Kim[3] to characterize the graphs with one hole and competition number one. We give a sufficient condition for a graph with one hole to have competition number one. This generates a huge class of graphs with one hole and competition number one. Then we completely characterize the graphs with one hole and competition number one that do not have a vertex adjacent to all the vertices of the hole. Also we show that deleting pendant vertices from a connected graph does not change the competition number of the original graph as long as the resulting graph is not trivial, and this allows us to construct infinitely many graph having the same competition number. Finally we pose an interesting open problem.
 Keywords
competition number;chordal graph;chordless cycle;hole;
 Language
English
 Cited by
1.
THE COMPETITION NUMBERS OF HAMMING GRAPHS WITH DIAMETER AT MOST THREE,;;

대한수학회지, 2011. vol.48. 4, pp.691-702 crossref(new window)
1.
Fuzzy k-competition graphs and p-competition fuzzy graphs, Fuzzy Information and Engineering, 2013, 5, 2, 191  crossref(new windwow)
2.
The competition number of a graph whose holes do not overlap much, Discrete Applied Mathematics, 2010, 158, 13, 1456  crossref(new windwow)
3.
An upper bound for the competition numbers of graphs, Discrete Applied Mathematics, 2010, 158, 2, 154  crossref(new windwow)
4.
$$m$$ m -Step fuzzy competition graphs, Journal of Applied Mathematics and Computing, 2015, 47, 1-2, 461  crossref(new windwow)
5.
The competition number of a graph with exactly two holes, Journal of Combinatorial Optimization, 2012, 23, 1, 1  crossref(new windwow)
6.
Graphs having many holes but with small competition numbers, Applied Mathematics Letters, 2011, 24, 8, 1331  crossref(new windwow)
7.
The competition number of a graph and the dimension of its hole space, Applied Mathematics Letters, 2012, 25, 3, 638  crossref(new windwow)
8.
Competition Numbers, Quasi-line Graphs, and Holes, SIAM Journal on Discrete Mathematics, 2014, 28, 1, 77  crossref(new windwow)
9.
A sufficient condition for Kim’s conjecture on the competition numbers of graphs, Discrete Mathematics, 2012, 312, 6, 1123  crossref(new windwow)
10.
THE COMPETITION NUMBERS OF HAMMING GRAPHS WITH DIAMETER AT MOST THREE, Journal of the Korean Mathematical Society, 2011, 48, 4, 691  crossref(new windwow)
11.
The competition number of a graph with exactly h holes, all of which are independent, Discrete Applied Mathematics, 2009, 157, 7, 1337  crossref(new windwow)
 References
1.
J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, North Holland, New York, 1976

2.
C. Cable, K. F. Jones, J. R. Lundgren, and S. Seager, Niche graphs, Discrete Appl. Math. 23 (1989), 231-241 crossref(new window)

3.
H. H. Cho and S-R. Kim, The Competition Number of a Graph Having Exactly One Hole, to appear in Discrete Math

4.
H. H. Cho, S-R. Kim, and Y. Nam, The m-Step Competition Graph of a Digraph, Discrete Appl. Math. 105 (2000), 115-127 crossref(new window)

5.
J. E. Cohen, Interval Graphs and Food Webs: A Finding and a Problem, RAND Corporation Document 17696-PR, Santa Monica, CA, 1968

6.
M. B. Cozzens and F. S. Roberts, T-Colorings of Graphs and the Channel As-signment Problem, Congr. Numer. 25 (1982) 191-208

7.
G. A. Dirac, On Rigid Circuit Graphs, Abh. Math. Sem. Univ. Hamburg 25 (1961), 71-76 crossref(new window)

8.
W. K. Hale, Frequency Assignment: Theory and Application, Proc. IEEE 68 (1980), 1497-1514 crossref(new window)

9.
P. C. Fishburn and W. V. Gehrlein, Niche numbers, J. Graph Theory 16 (1992), 131-139 crossref(new window)

10.
F. Harary, S-R. Kim, and F. S. Roberts, Competition numbers as a generalization of Turan's theorem, J. Ramanujan Math. Soc. 5 (1990), 33-43

11.
S-R. Kim, The Competition Number and Its Variants, in Quo Vadis, Graph Theory, (J. Gimbel, J. W. Kennedy, and L. V. Quintas, eds.), Annals of Discrete Mathematics 55, North Holland B. V., Amsterdam, the Netherlands, 1993, 313- 326

12.
S-R. Kim and F. S. Roberts, Competition numbers of graphs with a small number of triangles, Discrete Appl. Math. 78 (1997), 153-162 crossref(new window)

13.
J. R. Lundgren, Food Webs, Competition Graphs, Competition-Common Enemy Graphs, and Niche Graphs, in Applications of Combinatorics and Graph Theory to the Biological and Social Sciences, (F. S. Roberts, ed.), IMH Volumes in Mathematics and Its Application, Vol. 17, Springer-Verlag, New York, 1989, 221-243

14.
R. J. Opsut, On the Computation of the Competition Number of a Graph, SIAM J. Discrete Math. 3 (1982), 420-428 crossref(new window)

15.
R. J. Opsut and F. S. Roberts, On the Fleet Maintenance, Mobile Radio fre- quency, Task Assignment and Traffc phasing Problem, The Theory and Applications of Graphs, (G. Chartrand, Y. Alavi, D. L. Goldsmith, L. Lesniak-Foster, and D. R. Lick, eds.), Wiley, New York, 1981, 479-492

16.
F. S. Roberts, Food Webs, Competition Graphs, and the Boxicity of Ecological Phase Space, Theory and Applications of Graphs, (Y. Alavi and D. Lick, eds.), Springer Verlag, New York, 1978, 477-490

17.
F. S. Roberts, Graph Theory and Its Applications to Problems of Society, SIAM, Pennsylvania, 1978

18.
D. Scott, The competition-common enemy graph of a digraph, Discrete Appl. Math. 17 (1987), 269-280 crossref(new window)

19.
S. Seager, The Double Competition Number of Some Triangle-Free Graphs, Discrete Appl. Math. 29 (1990), 265-269

20.
C. E. Shannon, The Zero Capacity of a Noisy Channel, IEEE Trans. Inform. Theory IT-2 (1956), 8-19