An Efficient Algorithm to Compute Dot Product Dimension of Some Outerplanar Graphs

A graph G = (V (G),E(G)) is called a k-dot product graph if there is a function f : V (G) -> R-k such that for any two distinct vertices u and v, f(u).f(v) >= 1 if and only if uv is an element of E(G). The minimum value k such that G is a k-dot product graph, is called the dot product dimension rho(G) of G. In this paper, we give an efficient algorithm for computing the dot product dimension of outerplanar graphs of at most two edge-disjoint cycles. If the graph has two cycles, we only consider those outerplanar graphs if both cycles have exactly one vertex in common and the length of one of the cycles is greater than or equal to six.