TL;DR
This paper investigates graphs with exactly two trivial distance ideals, providing an infinite family of forbidden induced subgraphs and exploring their connections with other well-known graph classes.
Contribution
It introduces an infinite family of forbidden induced subgraphs for graphs with two trivial distance ideals and relates these graphs to other established classes.
Findings
Identifies an infinite family of forbidden subgraphs.
Characterizes graphs with two trivial distance ideals.
Links these graphs to other known graph classes.
Abstract
Distance ideals generalize the Smith normal form of the distance matrix of a graph. The family of graphs with 2 trivial distance ideals contains the family of graphs whose distance matrix has at most 2 invariant factors equal to 1. Here we give an infinite family of forbidden induced subgraphs for the graphs with 2 trivial distance ideals. These are also related with other well known graph classes.
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