Hamiltonian problems in directed graphs with simple row patterns

Abstract

We define a row pattern of a matrix as a model (or a condition) to be satisfied by each row of the matrix, independently of the other rows. When applied to adjacency matrices of digraphs, a row pattern allows to define a class of digraphs characterized by a local condition on the out-neighborhood of each vertex, once an appropriate linear ordering of the vertices has been found.We show that the Hamiltonian Cycle and Hamiltonian Path problems are NP-complete on digraphs with simple row patterns, namely C1P-digraphs (where the row pattern describes the consecutive ones property), 2-out-regular digraphs (where each row contains exactly two values 1) and sym-digraphs (where the 1s in each row are symmetric with respect to the main diagonal, as soon as the row contains at least two 1s).Classes with simple row patterns thus show a real structural complexity, despite their simple definitions. Moreover, they raise interesting class recognition problems.