1. In an ad-hoc wireless communications network comprised of a plurality of nodes, a method of establishing an arterial sub-network of nodes such that each of the plurality of nodes in the communications network is within one hop of a node in the arterial sub-network, the anterial sub-network of nodes assigning slots to neighboring nodes, the method comprising steps of: (a) assigning a first node of the plurality of nodes to be part of the arterial sub-network when a set comprising the one-hop neighbors of the first node is a proper superset of each of the first node's one-hop neighbors' sets of neighbors; the method further including assigning the first node to not be a part of the arterial sub-network when one of the following is true: (b) the set of one-hop neighbors of the first node is a proper subset of another node's one-hop neighbors; (c) the set of one-hop neighbors of the first node is equal to a set of one-hop neighbors of a second node, and an identifier of the first node is different from an identifier of the second node such that the identifiers prioritize the second node over the first node; (d) the first node has a set of neighbors that is different from at least one of said first node's neighbors' neighbor sets and said first node's neighbors are either directly connected or they are connected by a node having one-hop neighbors that are reporting disjoint sets of one-hop neighbors, (e) all nodes in the intersection of the neighbor sets of said first node's neighbors have the same number of neighbors where said first node's identifier is less than the identifier of one of said all nodes or one of said all nodes has more neighbors than said first node; (f) the set of one-hop neighbors of said each node is equal to a set of one-hop neighbors of another node, and the identifier of said each node is different from an identifier of said another node such that the identifiers prioritize said another node over said each node, (g) all nodes in the intersection of the neighbor sets of said each node's neighbors have the same number of neighbors where said each node's identifier is less than the identifier of one of said all nodes or one of said all nodes has more neighbors than said each node, and (h) said each node has a set of neighbors that is different from at least one of said each node's neighbors' neighbor sets and said each node's neighbors are either directly connected or they are connected by the node having one-hop neighbors that are reporting disjoint sets of one-hop neighbors, and further wherein said each node is assigned to be part of the arterial sub-network when none of conditions (a), (b), (c), (d), (e), (f), (g), or (h) are true and when one of said each node has no one-hop neighbors that are part of the arterial sub-network, and two one-hop neighbors of said each node each have a set of neighbors excluding said each node that are part of the arterial sub-network wherein said sets of neighbors have no nodes in common are true.