A One-Vertex Decomposition Algorithm for Generating Algebraic Expressions of Square Rhomboids

Abstract

The paper investigates relationship between algebraic expressions and graphs. We consider a digraph called a square rhomboid that is an example of non-seriesparallel graphs. Our intention is to simplify the expressions of square rhomboids and eventually find their shortest representations. With that end in view, we describe the new algorithm for generating square rhomboid expressions, which improves on our previous algorithms.

Full Square Rhomboids and Their Algebraic Expressions

“…The expressions of square rhomboids are generated using two-vertex decomposition method (2-VDM) and one-vertex decomposition method (1-VDM) [11]. Both methods are based on revealing subgraphs in the initial graph.…”

“…In [11] we considered a non-series-parallel st-dag called a square rhomboid (Figure 3). This graph looks like a planar approximation of the square of a rhomboid, which is a series composition of rhomb graphs.…”

“…Some algorithms which generate the expressions of O n log 2 6 complexity for SR(n) are discussed in [11]. Our intention in this paper is to generate and simplify the expressions of full square rhomboids.…”

“…However, numerically 1-VDM is more efficient than 2-VDM [11]. It follows from discussed in [11] explicit formulae for T (n) as well (n = 2 k for some positive integer k ≥ 2).…”

“…However, numerically 1-VDM is more efficient than 2-VDM [11]. It follows from discussed in [11] explicit formulae for T (n) as well (n = 2 k for some positive integer k ≥ 2). The coefficient of the leading term of the formulaen log 2 6 is equal to 212 135 ≈ 1.57 for the best algorithm based on 2-VDM and 154 135 ≈ 1.14 for 1-VDM.…”