Interval Routing and Minor-Monotone Graph Parameters

Abstract

We survey a number of minor-monotone graph parameters and their relationship to the complexity of routing on graphs. In particular we compare the interval routing parameters κslir(G) and κsir(G) with Colin de Verdi`ere’s graph invariant μ(G) and its variants λ(G) and κ(G). We show that for all the known characterizations of θ(G) with θ(G) being μ(G), λ(G) or κ(G), that θ(G) ≤ 2κslir(G) - 1 and θ(G) ≤ 2κsir(G) and conjecture that these inequalities always hold. We show that θ(G) ≤ 4κslir(G) - 1 and θ(G) ≤ κsir(G) + 1