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

Abstract

A graph [Formula: see text] is called a [Formula: see text]-dot product graph if there is a function [Formula: see text] such that for any two distinct vertices [Formula: see text] and [Formula: see text], [Formula: see text] if and only if [Formula: see text]. The minimum value [Formula: see text] such that [Formula: see text] is a [Formula: see text]-dot product graph, is called the dot product dimension [Formula: see text] of [Formula: see text]. 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.