Metric problems on graphs

Some properties of homometric sets are known in Music Theory for at least fifty years [16]. We define homometric sets in graphs as follows. Let G = (V,E) denote a simple graph with the vertex set V and the edge set E. Given a vertex set V ′ ⊆ V , the profile of V ′ denotes the multiset of pairwise distances in G between the vertices of V ′. Two disjoint vertex sets of the same size A ⊆ V and B ⊆ V are called homometric if they have identical profiles. In this thesis, we study lower bounds on the size of homometric sets in various classes of graphs. We prove that every tree on n vertices contains a homometric pair of sets of size at least √ n 2 − 1 2 , thereby improving the best previously known lower bound of n1/3 by Axenovich and Özkahya [3]. We also investigate diameter-two graphs and improve the result from the same paper in the case of sparse diameter-two graphs. In particular, we show that a diameter-two graph on n vertices and m edges contains homometric sets of size at least n 2 c1m , for a fixed constant c1 > 0, whereas Axenovich and Özkahya proved a lower bound of √ n < n 2 c1m , for m ∈ o ( n3/2 ) , for general diameter-two graphs. In the case of outerplanar graphs and planar 3-trees, we improve the best previously known lower bound of logn log logn implied by the result on general graphs by Albertson, Pach and Young [1]. More precisely, in those two classes of graphs we are able to find homometric sets of size at least c2 log n, for a fixed constant c2 > 0. If G is a ladder graph on n vertices, we show that its vertex set contains a pair of homometric sets of size at least c3 √ n, for a fixed constant c3 > 0.