Axiomatics for oriented connectivity

Abstract

Tankyevych et al. (2013) considered, in a directed graph, any set S of vertices where there is some marker vertex p ∈ S such that for every vertex x ∈ S , there is a directed path from p to x included in S ; the family of all such sets S was called a semi-connection. Their properties were briefly analysed and compared with connectivity and connected components in undirected graphs.We give an abstract algebraic formalization of this concept, following the same approach as that of Serra (1988) and Ronse ( 2008) for the notions of connection and partial connection, which generalize both topological and graph-theoretic connectivity.Here the sets S are unsufficient, one must associate to them their markers p; thus in a space E we consider a family R of ordered pairs (p, S ) ∈ E × P(E), where the set S can be "reached" from marker p; this family, which we call a reach, must satisfy the three properties of union, transitivity and membership; a fourth point property leads to a full reach. As in (Serra, 1988;Ronse, 2008), we give an equivalent definition in terms of a system of point openings (γ p , p ∈ E) satisfying some properties. The special case of symmetry, where S does not depend on the choice of the marker p ∈ S , leads to a partial connection or a connection. Some examples are given.Possible applications of this new theory lie in the analysis of connected structures having an orientation, for instance vascular networks in medical imaging. One can also apply it to geodesic reconstruction and connected filtering.

Oriented components and their separations

“…When generation rules apply to individual elements, as in the case of mutations of gene sequences, reaches in the sense of [28] and [22] are obtained. Generative grammars and (term) rewriting systems could also be viewed in this way.…”

“…Axiom (U) generalizes the "union property" of [22]. It allows multiple production to be "applied" at the same time.…”

“…It is shown e.g. in [22] that there is a bijection between the openings γ P : (P, Q) → Q[P ] and the set systems with the union property F P := {Q|P ≻ − Q} for fixed P . The theorem now follows directly from lemma 3.6 and lemma 3.8, whose proofs also establish the correspondence P ≻ − Q =⇒ P ′ ≻ − Q for all…”

“…Ronse explored this concept in more detail [22] as a generalization of connectivity openings [20,23] and discussed some fundamental properties of the equivalent notion of oriented components. The latter form a system of pairs (p, Q) with the interpretation that "every point of Q is reachable from p within Q".…”

“…The key point is that in a chemical reaction, the "output molecules" depend on a set of "input molecules" rather than on a single input molecule. This suggests to generalize the work of [28,22] to pairs (P, Q) such that every point of Q can be reached from the start set P within Q.…”