Perfect Digraphs

Abstract

The clique number ω(D) of a digraph D is the size of the largest bidirectionally complete subdigraph of D. D is perfect if, for any induced subdigraph H of D, the dichromatic number χ(H) defined by Neumann‐Lara (The dichromatic number of a digraph, J. Combin. Theory Ser. B 33 (1982), 265–270) equals the clique number ω(H). Using the Strong Perfect Graph Theorem (M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, The strong perfect graph theorem, Ann. Math. 164 (2006), 51–229) we give a characterization of perfect digraphs by a set of forbidden induced subdigraphs. Modifying a recent proof of Bang‐Jensen et al. (Finding an induced subdivision of a digraph, Theoret. Comput. Sci. 443 (2012), 10–24) we show that the recognition of perfect digraphs is co‐scriptNP‐complete. It turns out that perfect digraphs are exactly the complements of clique‐acyclic superorientations of perfect graphs. Thus, we obtain as a corollary that complements of perfect digraphs have a kernel, using a result of Boros and Gurvich (Perfect graphs are kernel solvable, Discrete Math. 159 (1996), 35–55). Finally, we prove that it is NP‐complete to decide whether a perfect digraph has a kernel.

Topological Interference Management With Decoded Message Passing

“…For a perfect digraph D, acyclic set coloring of D reduces to vertex coloring of its symmetric part S(D). Thus, any feasible coloring of the symmetric graph S(D) is also feasible for D [26]. If the conflict digraph D only has symmetric part, then χ A (D) = χ(S(D)).…”

“…Interested readers are suggested referring to the text books [25], [36] and papers [21], [26] for more details.…”

Segmenting Planar Superpixel Adjacency Graphs w.r.t. Non-planar Superpixel Affinity Graphs

“…Notice also that the optimal value of each cycle sub-problem is zero so the term corresponding to them in (3) can be removed. 1 Notice that we impose a requirement on θ pmc not a requirement on x pmc . 2 A 2-colorable segmentation is a partition of the set of superpixels into connected subsets (segments) such that every segment can be given a different color from all adjacent segments, using only two colors overall.…”

“…We uncut all cut edges within connected components. To select the optimal T we can try all unique values of Z T γ or a fixed number of values uniformly spaced over the interval [0,1]. Notice that Z T γ ≥ T defines which non-planar edges are cut.…”

“…Both, our implementation of Alg. 1 and the optimized C++ implementation [1] of [3], use IBM ILOG Cplex as an LP and ILP solver, respectively. We use Blossom V [11] for computing the optimal 2-colorable partitions.…”