Some results on tree tolerance representations

Mary Ann Saadi, University of Rhode Island


Consider a simple undirected graph G = ( V, E). A family of subtrees, {Sv} v∈V, of a host tree H is called a t-subtree representation of G provided uv E if and only if :V(Su) ∩ V(Sv): ≥ t. An ( H, t)-representation of G is a t-subtree representation of G with fixed host tree H. In this dissertation we will consider (Hm, t)-representations for cycles and trees, where Hm is the host tree with exactly one node v, of degree three and exactly three leaves, each with distance m from v. We denote the set of (Hm, t)-representable graphs for some positive integer m, as H (t). We derive an upper and lower bound for the maximum size of a cycle in H (t). Then we characterize the set of all trees in H (t) for t = 1, 2 and 3. In addition, we determine a necessary condition for a graph G to be in H (1) and conjecture that this condition is also sufficient. In our last result we show that any graph G in H (t) is also in H (t + 1) for all t. ^

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Recommended Citation

Mary Ann Saadi, "Some results on tree tolerance representations" (2001). Dissertations and Master's Theses (Campus Access). Paper AAI3039083.