# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

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# Discussiones Mathematicae Graph Theory

## Weak Saturation Numbers for Sparse Graphs

 Ralph J. Faudree Department of Mathematical Sciences University of Memphis Memphis, TN 38152 Ronald J. Gould Department of Math and Computer Science Emory University Atlanta, GA 30322 Michael S. Jacobson Department of Mathematical and Statistical Sciences University of Colorado Denver Denver, CO 80217

## Abstract

For a fixed graph F, a graph G is F-saturated if there is no copy of F in G, but for any edge e ∉ G, there is a copy of F in G + e. The minimum number of edges in an F-saturated graph of order n will be denoted by sat(n, F). A graph G is weakly F-saturated if there is an ordering of the missing edges of G so that if they are added one at a time, each edge added creates a new copy of F. The minimum size of a weakly F-saturated graph G of order n will be denoted by wsat(n, F). The graphs of order n that are weakly F-saturated will be denoted by wSAT(n, F), and those graphs in wSAT(n, F) with wsat(n, F) edges will be denoted by wSAT(n,F). The precise value of wsat(n,T) for many families of sparse graphs, and in particular for many trees, will be determined. More specifically, families of trees for which wsat(n, T) = |T | − 2 will be determined. The maximum and minimum values of wsat(n,T) for the class of all trees will be given. Some properties of wsat(n,T) and wSAT(n,T) for trees will be discussed.

Keywords: saturated graphs, sparse graphs, weak saturation

2010 Mathematics Subject Classification: 05C35.

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