Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 4 1 2019 06 01 Strong Alliances in Graphs 1 13 13785 10.22049/cco.2018.25921.1056 EN C. Hegde Mangalore University B. Sooryanarayana Dr. Ambedkar Institute of Technology Journal Article 2017 04 09 For any simple connected graph $G=(V,E)$, a defensive alliance is a subset $S$ of $V$ satisfying the condition that every vertex $v\in S$ has at most one more neighbour in $V-S$ than it has in $S$. The minimum cardinality of any defensive alliance in $G$ is called the alliance number of $G$, denoted $a(G)$. In this paper, we introduce a new type of alliance number called $k$-strong alliance number and its varieties. The bounds for 1-strong alliance number in terms of different graphical parameters are determined and the characterizations of graphs with 1-strong alliance number 1, 2, and $n$ are obtained. https://comb-opt.azaruniv.ac.ir/article_13785_db97a57dfa7d3980c88f4ce8245a31b6.pdf