Exploring structural properties of $k$-trees and block graphs

TL;DR

This paper introduces a new characterization of $k$-trees using reduced clique graphs and $(k+1)$-line graphs, revealing a relationship with spanning trees of block graphs, and provides a linear-time method for counting spanning trees.

Contribution

It presents a novel characterization of $k$-trees and block graphs, establishing a link between clique-trees and spanning trees, with an efficient counting approach.

Findings

Number of clique-trees of a $k$-tree equals the number of spanning trees of its $(k+1)$-line graph.

The relationship enables a linear-time algorithm for counting spanning trees in block graphs.

Structural properties of $k$-trees and block graphs are characterized through reduced clique graphs.

Abstract

We present a new characterization of $k$-trees based on their reduced clique graphs and $(k+1)$-line graphs, which are block graphs. We explore structural properties of these two classes, showing that the number of clique-trees of a $k$-tree $G$ equals the number of spanning trees of the $(k+1)$-line graph of $G$. This relationship allows to present a new approach for determining the number of spanning trees of any connected block graph. We show that these results can be accomplished in linear time complexity.