Totally nonpositive completions on partial matrices

An n × n real matrix is said to be totally nonpositive if every minor is nonpositive. In this paper, we are interested in totally nonpositive completion problems, that is, does a partial totally nonpositive matrix have a totally nonpositive matrix completion? This problem has, in general, a negative answer. Therefore, we analyze the question: for which labeled graphs G does every partial totally nonpositive matrix, whose associated graph is G, have a totally nonpositive completion? Here we study the mentioned problem when G is a chordal graph or an undirected cycle.