What is the reason for calling independent sets as stable sets of a graph? Is the stable set says anything more about the graph other than it is just a set of pairwise non adjacent vertices?
2026-03-27 10:44:27.1774608267
why the name stable set?
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"Stable" sets used to be called "internally stable" sets, and the independence number $\alpha(G)$ used to be called the "coefficient of internal stability". Unfortunately, none of the early adopters explain why this makes any sense.
The earliest occurrence of "internally stable" sets that I could find was in Berge's article Two theorems in graph theory (1957), and later in his 1958 textbook on graph theory. It's possible that in French, there's an obvious connection between "internal stability" and vertices not being adjacent that justifies the terminology.
I can also find some references (e.g., this paper) which refer to a dominating set (that is, a set $S$ such that every vertex in $V(G) \setminus S$ has a neighbor in $S$) as an "externally stable" set.