Blazing a Trail via Matrix Multiplications: A Faster Algorithm for Non-shortest Induced Paths

Abstract

For vertices u and v of an n-vertex graph G, a uv-trail of G is an induced uv-path of G that is not a shortest uv-path of G. Berger, Seymour, and Spirkl gave the previously only known polynomialtime algorithm, running in O(n 18 ) time, to either output a uv-trail of G or ensure that G admits no uv-trail. We reduce the complexity to the time required to perform a poly-logarithmic number of multiplications of n 2 × n 2 Boolean matrices, leading to a largely improved O(n 4.75 )-time algorithm.

When all holes have the same length

“…)-free and is therefore a threshold graph. This proves (3). By the definition of odd pretemplates, x is in the interior of a path P = v i .…”

“…Hence from now on, we may assume that u a = u (hence a is not adjacent to u) and that if u a = u + then u a = u a . Now by (3) we get that S • = ∅.…”

“…Now the sets K v for all v ∈ (A ∪ A ∪ I) \ {u}, K u ∪ {x}, B * and B * form a preblowup of F 0 . All conditions are easily checked, in particular K u ∪ {x} is a clique by (1), it satisfies condition (i) by (2) and condition (i1) by (3).…”

“…Let xy be an edge of G[B]. Up to symmetry, we may assume that N A (x) ⊆ N A (y), for otherwise vertices v i ∈ N A (x) \ N A (y) and v j ∈ N A (y) \ N A (x) either form a C 4 with x and y or a contradiction to (3).…”

“…In [1] and [3], polynomial-time algorithms that, given a graph G and u, v ∈ V (G), decide whether there exists a path from u to v that is not a shortest path are described. It is easy to deduce from such algorithms an algorithm to recognize C k in polynomial time.…”