Fully Polynomial-time Algorithms Parameterized by Vertex Integrity Using Fast Matrix Multiplication

We study the computational complexity of several polynomial-time-solvable graph problems parameterized by vertex integrity, a measure of a graph's vulnerability to vertex removal in terms of connectivity. Vertex integrity is the smallest number $\iota$ such that there is a set $S$ of $\iota' \le \iota$ vertices such that every connected component of $G-S$ contains at most $\iota-\iota'$ vertices. It is known that the vertex integrity lies between the well-studied parameters vertex cover number and tree-depth. Alon and Yuster [ESA 2007] designed algorithms for graphs with small vertex cover number using fast matrix multiplications. We demonstrate that fast matrix multiplication can also be effectively used when parameterizing by vertex integrity $\iota$ by developing efficient algorithms for problems including an $O(\iota^{\omega-1}n)$-time algorithm for computing the girth of a graph, randomized $O(\iota^{\omega - 1}n)$-time algorithms for Maximum Matching and for finding any induced four-vertex subgraph except for a clique or an independent set, and an $O(\iota^{(\omega-1)/2}n^2) \subseteq O(\iota^{0.687} n^2)$-time algorithm for All-Pairs Shortest Paths. These algorithms can be faster than previous algorithms parameterized by tree-depth, for which fast matrix multiplication is not known to be effective.