TY - JOUR

T1 - On the number of update digraphs and its relation with the feedback arc sets and tournaments

AU - Aracena, J.

AU - Demongeot, J.

AU - Fanchon, E.

AU - Montalva, M.

N1 - Funding Information:
This work was partially supported by FONDECYT project 1090549 (J.A., M.M.) and ANILLO ACT-87 (J.A.), by FONDAP and BASAL projects CMM, Universidad de Chile, by Centro de Investigación en Ingeniería Matemática (), Universidad de Concepción, by CONICYT fellowship, MECESUP Project UCO0713 and ANILLO ACT-88 (M.M.). Besides, we gratefully thank Andras Sebö for bringing tournaments to our attention.

PY - 2013/7

Y1 - 2013/7

N2 - An update digraph corresponds to a labeled digraph that indicates a relative order of its nodes introduced to define equivalence classes of deterministic update schedules yielding the same dynamical behavior of a Boolean network. In Aracena et al. [1], the authors exhibited relationships between update digraphs and the feedback arc sets of a given digraph G. In this paper, we delve into the study of these relations. Specifically, we show differences and similarities between both sets through increasing and decreasing monotony properties in terms of their structural characteristics. Besides, we prove that these sets are equivalent if and only if all the digraph circuits are cycles. On the other hand, we characterize the minimal feedback arc sets of a given digraph in terms of their associated update digraphs. In particular, for complete digraphs, this characterization shows a close relation with acyclic tournaments. For the latter, we show that the size of the associated equivalence classes is a power of two. Finally, we determine exactly the number of update digraphs associated to digraphs containing a tournament.

AB - An update digraph corresponds to a labeled digraph that indicates a relative order of its nodes introduced to define equivalence classes of deterministic update schedules yielding the same dynamical behavior of a Boolean network. In Aracena et al. [1], the authors exhibited relationships between update digraphs and the feedback arc sets of a given digraph G. In this paper, we delve into the study of these relations. Specifically, we show differences and similarities between both sets through increasing and decreasing monotony properties in terms of their structural characteristics. Besides, we prove that these sets are equivalent if and only if all the digraph circuits are cycles. On the other hand, we characterize the minimal feedback arc sets of a given digraph in terms of their associated update digraphs. In particular, for complete digraphs, this characterization shows a close relation with acyclic tournaments. For the latter, we show that the size of the associated equivalence classes is a power of two. Finally, we determine exactly the number of update digraphs associated to digraphs containing a tournament.

KW - Feedback arc set

KW - Tournament

KW - Update digraph

KW - Update schedule

UR - http://www.scopus.com/inward/record.url?scp=84876415478&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2012.12.018

DO - 10.1016/j.dam.2012.12.018

M3 - Article

AN - SCOPUS:84876415478

VL - 161

SP - 1345

EP - 1355

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

IS - 10-11

ER -