Abstract
In this article we deal with the problems of finding the disimplicial arcs of a digraph and recognizing some interesting graph classes defined by their existence. A <i>diclique</i> of a digraph is a pair $V$ &rarr; $W$ of sets of vertices such that $v$ &rarr; $w$ is an arc for every $v$ &isin; $V$ and $w$ &isin; $W$. An arc $v$ &rarr; $w$ is <i>disimplicial</i> when it belongs to a unique maximal diclique. We show that the problem of finding the disimplicial arcs is equivalent, in terms of time and space complexity, to that of locating the transitive vertices. As a result, an efficient algorithm to find the bisimplicial edges of bipartite graphs is obtained. Then, we develop simple algorithms to build disimplicial elimination schemes, which can be used to generate bisimplicial elimination schemes for bipartite graphs. Finally, we study two classes related to perfect disimplicial elimination digraphs, namely weakly diclique irreducible digraphs and diclique irreducible digraphs. The former class is associated to finite posets, while the latter corresponds to dedekind complete finite posets.
Highlights
Disimplicial arcs are important when Gaussian elimination is performed on a sparse matrix, as they correspond to the entries that preserve zeros when chosen as pivots
Since Split(D∗) can be computed in linear time when D is provided as input, we conclude that finding the disimplicial arcs of an ST graph is at least as hard as testing if a digraph is transitive
Since Split, Join, and Repr can be computed in linear time, we conclude that listing the disimplicial arcs and finding the transitive vertices are hard problems
Summary
Disimplicial arcs are important when Gaussian elimination is performed on a sparse matrix, as they correspond to the entries that preserve zeros when chosen as pivots. Any permutation of E is a perfect elimination scheme, the pivots of the matrix associated to G can be taken in any order from E with zero fill-in How to answer this question efficiently is already known, as it reduces to establishing if the thin arcs form a perfect matching of disimplicial arcs (see [2] and Section 3). An O(min{α, τ }m) time and O(m) space algorithm for a digraph with τ thin arcs is obtained, improving over the algorithm in [2] This algorithm is optimal unless an o(αm) time algorithm for finding the transitive vertices of a sparse graph is obtained, which is an open problem [11]. For the case in which all the arcs of the elimination scheme must belong to an input matching, we develop an O(αm) time and O(m) space; which is a major improvement for sparse graphs.
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