localMarkov -- local Markov statements for a graph or a directed graph

Given an undirected graph $G$, a local Markov statement is of the form \{$v$, non-neighbours($v$), neighbours($v$)\} . That is, every vertex $v$ of $G$ is independent of its non-neighbours given its neighbours.

For example, for the undirected 5-cycle graph $G$, that is, the graph on 5 vertices with $a—b—c—d—e—a$, we get the following local Markov statements:

Given a directed acyclic graph $G$, local Markov statements are of the form \{$v$, nondescendents($v$) - parents($v$), parents($v$)\} . In other words, every vertex $v$ of $G$ is independent of its nondescendents (excluding parents) given its parents.

For example, given the digraph $D$ on $7$ vertices with edges $1 \to 2, 1 \to 3, 2 \to 4, 2 \to 5, 3 \to 5, 3 \to 6, 4 \to 7, 5 \to 7$, and $6\to 7$, we get the following local Markov statements:

This method displays only non-redundant statements. In general, given a set $S$ of conditional independent statements and a statement $s$, then we say that $s$ is a a redundant statement if $s$ can be obtained from the statements in $S$ using the semigraphoid axioms of conditional independence: symmetry, decomposition, weak union, and contraction as described in Section 1.1 of Judea Pearl, Causality: models, reasoning, and inference, Cambridge University Press. We do not use the intersection axiom since it is only valid for strictly positive probability distributions.