Total Dual Integrality of Triangle Covering

This paper concerns weighted triangle covering in undirected graph \(G=(V,E)\), where a nonnegative integral vector \(\mathbf w=(w(e):e\in E)^T\) gives weights of edges. A subset S of E is a triangle cover in G if S intersects every triangle of G. The weight of a triangle cover is the sum of w(e) over all edges e in it. The characteristic vector \(\mathbf x\) of each triangle cover in G is an integral solution of the linear system where A is the triangle-edge incidence matrix of G. System \(\pi \) is totally dual integral if \(\max \{\mathbf 1^T\mathbf y:A^{T}\mathbf y\le \mathbf w,\mathbf y\ge \mathbf 0\}\) has an integral optimum solution \(\mathbf y\) for each integral vector \(\mathbf w\in \mathbb Z_+^E\) for which the maximum is finite. The total dual integrality of \(\pi \) implies the nice combinatorial min-max relation that the minimum weight of a triangle cover equals the maximize size of a triangle packing, i.e., a collection of triangles in G (repetitions allowed) such that each edge e is contained in at most w(e) of them. In this paper, we obtain graphical properties that are necessary for the total dual integrality of system \(\pi \), as well as those for the (stronger) total unimodularity of matrix A and the (weaker) integrality of polyhedron \(\{\mathbf x:A\mathbf x\ge \mathbf 1,\mathbf x\ge \mathbf 0\}\). These necessary conditions are shown to be sufficient when restricted to planar graphs. We prove that the three notions of integrality coincide, and are commonly characterized by excluding odd pseudo-wheels from the planar graphs.