Uniquely Hamiltonian Characterizations of Distance-Hereditary and Parity Graphs

A graph is shown to be distance-hereditary if and only if no induced subgraph of order five or more has a unique hamiltonian cycle; this is also equivalent to every induced subgraph of order five or more having an even number of hamiltonian cycles. Restricting the induced subgraphs to those of odd order five or more gives two similar characterizations of parity graphs. The close relationship between distancehereditary and parity graphs is unsurprising, but their connection with hamiltonian cycles of induced subgraphs is unexpected. 1 Distance-hereditary graphs Howorka [10] defined a graph to be a distance-hereditary graph if the distance between vertices in connected induced subgraphs always equals the distance between them in the full graph. This is equivalent to every cycle of length five or more having two crossed chords; [1, 4, 7, 9] contain many additional characterizations. Lemma 1 will state two characterizations of distance-hereditary graphs from [1] that will be used below. As in [4], the vertices in a set S ⊂ V (G) are twins if they all have exactly the same neighbors in G − S and are either pairwise adjacent (in which case they are true twins) or pairwise nonadjacent (in which case they are false twins). A pendant vertex is a vertex v for which there is a vertex w such that N(v) = {w}; the edge vw is then a pendant edge. A graph G is a one-vertex expansion [1] of a graph G if V (G) = V (G) ∪ {v} 6= V (G) where either v has a twin vertex v in G or v is a pendant vertex of G. Lemma 1 (Bandelt & Muller 1986) Each of the following is equivalent to G being a distance-hereditary graph: the electronic journal of combinatorics 15 (2008), #N36 1 (1.1) Each component of G is built from a single vertex by a sequence of one-vertex expansions. (1.2) G contains no induced cycle of length five or more and no induced house, gem, or domino subgraph (see Figure 1).