Index of Parameters of Iterated Line Graphs

TL;DR

This paper investigates how fifteen graph parameters behave under iterated line-graph operations on prolific graphs, establishing unboundedness and exact indices for most parameters, with some open problems remaining.

Contribution

It determines the index of fifteen graph parameters over prolific graphs under iterated line-graph operations, providing exact values for most and characterizing extremal graphs.

Findings

All fifteen parameters are unbounded under iteration.

Exact indices are determined for twelve parameters.

Open problems remain for three parameters: independence number, independent domination number, and domination number.

Abstract

Let $G$ be a prolific graph, that is a finite connected simple graph which is not isomorphic to a cycle nor a path nor the star graph $K_{1,3}$. The line-graph of $G$, denoted by $L(G)$, is defined by having its vertex-set equal to the edge-set of $G$ and two vertices of $L(G)$ are adjacent if the corresponding edges are adjacent in $G$. For a positive integer $k$, the iterated line-graph $L^k(G)$ is defined recursively by $L^k(G)=L(L^{k-1}(G))$. We consider fifteen graph parameters and study their behaviour when the operation of taking the line-graph is iterated. We shall first show that all of these parameters are unbounded. This idea is motivated by a well-known result of van Rooij and Wilf that says that the number of vertices is unbounded if and only if the graph is prolific. We then study of the value of $k(P,{\mathcal F})$, which is the index of a family of prolific graphs with regards to a given graph parameter $P(G)$. For a given parameter $P(G)$, the index of $G$ is denoted by $ind(P,G) = \min \{ r : P(G) < P(L^r(G) \}$. Now for a family $\mathcal F$ of prolific graphs, the index of the family is $k(P,\mathcal{F}) = \max \{ ind(P,G) : G \in \mathcal F\}$. The problem of determining the index of a parameter over the family of prolific graphs is motivated by a result of Chartrand who showed that it could require $k=|V(G)|-3$ iterations for $L^k(G)$ to have a hamiltonian cycle. For twelve of the parameters considered, we exactly determine $k(P,\mathcal F)$ where $\mathcal F$ is the family of all prolific graphs, and for some parameters we also characterize the class of prolific graphs realizing the extremal value $k(P,\mathcal F$). Interesting open problems remain, in particular completing the determination of $k(P,\mathcal F)$ for the three parameters: the independence number, independent domination number and domination number.