Domination and independence subdivision numbers of graphs

• Autorzy: Teresa W. Haynes , Sandra M. Hedetniemi , Stephen T. Hedetniemi
• Języki publikacji: EN

Abstrakty

EN

The domination subdivision number $sd_γ(G)$ of a graph is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number. Arumugam showed that this number is at most three for any tree, and conjectured that the upper bound of three holds for any graph. Although we do not prove this interesting conjecture, we give an upper bound for the domination subdivision number for any graph G in terms of the minimum degrees of adjacent vertices in G. We then define the independence subdivision number $sd_β(G)$ to equal the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the independence number. We show that for any graph G of order n ≥ 2, either $G = K_{1,m}$ and $sd_β(G) = m$, or $1 ≤ sd_β(G) ≤ 2$. We also characterize the graphs G for which $sd_β(G) = 2$.

Słowa kluczowe

EN

domination; independence; subdivision numbers

Kategorie tematyczne

• 05C69: Dominating sets, independent sets, cliques

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2000
• Tom: 20
• Numer: 2
• Strony: 271-280

Daty

wydano

2000

otrzymano

2000-08-04

Twórcy

autor

Teresa W. Haynes

• Department of Mathematics, East Tennessee State University, Johnson City, TN 37614 USA

autor

Sandra M. Hedetniemi

• Department of Computer Science, Clemson University, Clemson, SC 29634 USA

autor

Stephen T. Hedetniemi

• Department of Computer Science, Clemson University, Clemson, SC 29634 USA

Bibliografia

• [1] S. Arumugam, private communication, June 2000.
• [2] C. Berge, Graphs and Hypergraphs (North-Holland, Amsterdam, 1973).
• [3] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (American Elsevier, New York, 1977).
• [4] G. Chartrand and L. Lesniak, Graphs & Digraphs (Wadsworth and Brooks/Cole, Monterey, CA, third edition, 1996).
• [5] F. Harary, Graph Theory (Addison-Wesley, Reading, MA, 1969).
• [6] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, 1998).
• [7] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs, Advanced Topics (Marcel Dekker, Inc., New York, 1998).
• [8] G. Hopkins and W. Staton, Graphs with unique maximum independent sets, Discrete Math. 57 (1985) 245-251, doi: 10.1016/0012-365X(85)90177-3.
• [9] D.B. West, Introduction to Graph Theory (Prentice Hall, New Jersey, 1996).

Identyfikatory

DOI

10.7151/dmgt.1126