Exact Tie-Breaking and Tight Degree Bounds for Minimum-Degree Greedy on Interval Graphs

Abstract

Minimum-degree greedy repeatedly selects a vertex of minimum current degree and deletes its closed neighborhood; let $B(G)$ and $W(G)$ be its largest and smallest output sizes over complete legal executions. Given a closed interval representation of an $n$-vertex graph, we compute both values and construct attaining executions in $O(n^3)$ time and $O(n^2)$ space using endpoint-window dynamic programming and legal interleaving. For interval graphs with $Δ(G)\le4$, we prove the tight bound $B(G)\ge3α(G)/4$ by reduction to unit interval graphs and induction with virtual boundary vertices. A connected unit interval family $J_t$ attains equality, with $B(J_t)=W(J_t)=3t$ and $α(J_t)=4t$. For each degree bound $d\ge5$, a degree-five family $H_t$ approaches the known $2/3$ lower bound, giving infimum $2/3$. Together with known optimal favorable tie-breaking for $d\le3$, these results determine the favorable guarantee for every degree bound.