Weak sense of direction labelings and graph embeddings

• Authors: Christine T. Cheng;Ichiro Suzuki
• Venue: Discrete Applied Mathematics
• Year: 2011

An edge-labeling @l for a directed graph G has a weak sense of direction (WSD) if there is a function f that satisfies the condition that for any node u and for any two label sequences @a and @a^' generated by non-trivial walks on G starting at u, f(@a)=f(@a^') if and only if the two walks end at the same node. The function f is referred to as a coding function of @l. The weak sense of direction number of G, WSD(G), is the smallest integer k so that G has a WSD-labeling that uses k labels. It is known that WSD(G)=@D^+(G), where @D^+(G) is the maximum outdegree of G. Let us say that a function @t:V(G)-V(H) is an embedding from G onto H if @t demonstrates that G is isomorphic to a subgraph of H. We show that there are deep connections between WSD-labelings and graph embeddings. First, we prove that when f"H is the coding function that naturally accompanies a Cayley graph H and G has a node that can reach every other node in the graph, then G has a WSD-labeling that has f"H as a coding function if and only if G can be embedded onto H. Additionally, we show that the problem ''Given G, does G have a WSD-labeling that uses a particular coding function f?'' is NP-complete even when G and f are fairly simple. Second, when D is a distributive lattice, H(D) is its Hasse diagram and G(D) is its cover graph, then WSD(H(D))=@D^+(H(D))=d^*, where d^* is the smallest integer d so that H(D) can be embedded onto the d-dimensional mesh. Along the way, we also prove that the isometric dimension of G(D) is its diameter, and the lattice dimension of G(D) is @D^+(H(D)). Our WSD-labelings are poset-based, making use of Birkhoff's characterization of distributive lattices and Dilworth's theorem for posets.