# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2018-2019): c. 84%

# Discussiones Mathematicae Graph Theory

## ON STRATIFICATION AND DOMINATION IN GRAPHS

 Ralucca Gera Department of Applied Mathematics Naval Postgradute School Monterey, CA 93943-5216, USA e-mail: RGera@nps.edu Ping Zhang Department of Mathematics Western Michigan University Kalamazoo, MI 49008, USA e-mail: ping.zhang@wmich.edu

## Abstract

A graph G is 2-stratified if its vertex set is partitioned into two classes (each of which is a stratum or a color class), where the vertices in one class are colored red and those in the other class are colored blue. Let F be a 2-stratified graph rooted at some blue vertex v. An F-coloring of a graph is a red-blue coloring of the vertices of G in which every blue vertex v belongs to a copy of F rooted at v. The F-domination number γF(G) is the minimum number of red vertices in an F-coloring of G. In this paper, we study F-domination, where F is a 2-stratified red-blue-blue path of order 3 rooted at a blue end-vertex. We present characterizations of connected graphs of order n with F-domination number n or 1 and establish several realization results on F-domination number and other domination parameters.

Keywords: stratified graph, F-domination, domination.

2000 Mathematics Subject Classification: 05C15, 05C69.

## References

 [1] B. Bollobas and E.J. Cockayne, The irredundance number and maximum degree of a graph, Discrete. Math. 49 (1984) 197-199, doi: 10.1016/0012-365X(84)90118-3. [2] G. Chartrand, H. Gavlas, M.A. Henning and R. Rashidi, Stratidistance in stratified graphs, Math. Bohem. 122 (1997) 337-347. [3] G. Chartrand, T.W. Haynes, M.A. Henning and P. Zhang, Stratification and domination in graphs, Discrete Math. 272 (2003) 171-185, doi: 10.1016/S0012-365X(03)00078-5. [4] G. Chartrand, T.W. Haynes, M.A. Henning and P. Zhang, Stratified claw domination in prisms, J. Combin. Math. Combin. Comput. 33 (2000) 81-96. [5] G. Chartrand, L. Holley, R. Rashidi and N.A. Sherwani, Distance in stratified graphs, Czech. Math. J. 125 (2000) 135-146. [6] G. Chartrand and P. Zhang, Introduction to Graph Theory (McGraw-Hill, Boston, 2005). [7] E.J. Cockayne, R.M. Dawes and S.T. Hedetniemi, Total domination in graphs, Networks 10 (1980) 211-219, doi: 10.1002/net.3230100304. [8] G.S. Domke, J.H. Hattingh, S.T. Hedetniemi, R.C. Laskar and L.R. Markus, Restrained domination, preprint. [9] J.F. Fink and M.S. Jacobson, n-Domination in graphs, in: Y. Alavi and A.J. Schwenk, eds, Graph Theory with Applications to Algorithms and Computer Science, 283-300 (Kalamazoo, MI 1984), Wiley, New York, 1985. [10] R. Rashidi, The Theory and Applications of Stratified Graphs (Ph.D. Dissertation, Western Michigan University, 1994).