Complexity of circuit intersection in graphs Original Research Article

Consider the decision problem STRICT BOUNDED CIRCUIT INTERSECTION (SBCI): Given a finite graph G=(V, E) and K∈Z+. Is there a subset E′ ⊆ E with |E′| ⩾ K such that | E′ ∩ C | < | C |/L for every circuit C of G? The problem NONSTRICT BCI (NBCI) is the same, except that | E′ ∩ C | < | C |/L is replaced by | E′ ∩ C | ⩽ | C |/L. It is proved that SBCI is NP-complete for every fixed integer L ⩾2 even if G is planar and bipartite, and NBCI is NP-complete for every fixed integer L ⩾ 3 even if G is planar. For every fixed integer L ⩾ 4, NBCI is NP-complete even if G is planar and bipartite. The case L = 2 of SBCI answers a question of Welsh, stemming from knot theory. The case L = 2 of NBCI, motivated by coding theory, has been shown to be polynomial for every graph by Frank.