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原文摘要

We examine the computational complexity of well-known problems on subgraphs of the strong product of two paths. Namely, for Max Cut we show NP-hardness on induced subgraphs of $P \boxtimes P$ answering a question by Dvořák, Huynh, Joret, Liu, and Wood (2021). For 3-Edge Coloring we show NP-hardness on subgraphs of $P \boxtimes P$ and that it is equivalent to 3-Edge Coloring on planar graphs for induced subgraphs of $P \boxtimes P$. It is a long standing open question whether 3-Edge Coloring on planar graphs is NP-hard. As a second contribution, we consider graphs of the form $G \boxtimes P$ and show that Feedback Vertex Set on these graphs is FPT in the clique-width of $G$. Furthermore, we show that it can be solved in polynomial time if $G$ is chordal or co-comparable and that the computational complexity depends on the parity of $P$ if $G$ is bipartite. Namely, it is solvable in polynomial time if $P$ has an even number of vertices but is NP-complete otherwise. For Max Cut, we show that it can be solved in polynomial time if $G$ is bipartite.