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Frederick Kramer 2022-07-11 Answered
What classes of graphs result from T ¯ ?
I need help in characterizing the classes of graphs that results from taking the complementary of a tree, i.e., the graph that results from removing the edges of a tree from a complete graph. More formally, let T = ( V , E ) be an n-vertex tree with vertex set V and edge set E. Are there known results on the classes of graphs defined by T ¯ ?
There are two trivial cases. If T = S n , i.e., is a star tree (one single vertex has degree n 1 and the other vertices have degree 1), we have that S n ¯ = { v } K n 1 , i.e., S n ¯ is a graph with an isolate vertex (degree 0) and a ( n 1 ) vertex clique. I've got the feeling that P n ¯ (where P n is an n-vertex path graph) has a precise characterization but I can't put a name to it.
If there are no (or few) results about general T, can we say something about T ¯ if T belongs to a class of trees, for example caterpillar trees (trees in which the removal of all leaves produces a path graph), lobster trees (trees in which the removal of all leaves produces a caterpillar tree), ...?
Any help will be appreciated. Thank you.
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Answers (1)

Alexzander Bowman
Answered 2022-07-12 Author has 19 answers
Explanation:
It is not completely clear what kind of characterization you are looking for. Here's one: complements of trees are the maximal graphs not containing a complete bipartite graph as a spanning subgraph. That is, maximal graphs that do not contain a subgraph of the form K s , t , where s + t is the number of vertices of the graph. Indeed, a graph is connected if and only if its complement does not contain a subgraph of this form. Since trees are the minimal connected graphs, their complements are the maximal graphs with this property.

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