Distance-hereditary Graph - Definition and Characterization

Definition and Characterization

The original definition of a distance-hereditary graph is a graph G such that, if any two vertices u and v belong to a connected induced subgraph H of G, then some shortest path connecting u and v in G must be a subgraph of H, so that the distance between u and v in H is the same as the distance in G.

Distance-hereditary graphs may also be characterized in several other equivalent ways:

• They are the graphs in which every induced path is a shortest path, or equivalently the graphs in which every non-shortest path has at least one edge connecting two non-consecutive path vertices.

• They are the graphs in which every cycle of length at least five has two or more diagonals, and in which every cycle of length exactly five has at least one pair of crossing diagonals.

• They are the graphs in which every cycle of length five or more has at least one pair of crossing diagonals.

• They are the graphs in which, for every four vertices u, v, w, and x, at least two of the three sums of distances d(u,v)+d(w,x), d(u,w)+d(v,x), and d(u,x)+d(v,w) are equal to each other.

• They are the graphs that do not have as isometric subgraphs any cycle of length five or more, or any of three other graphs: a 5-cycle with one chord, a 5-cycle with two non-crossing chords, and a 6-cycle with a chord connecting opposite vertices.

• They are the graphs that may be built up from a single vertex by a sequence of the following three operations, as shown in the illustration:
  1. Add a new pendant vertex connected by a single edge to an existing vertex of the graph.
  2. Replace any vertex of the graph by a pair of vertices, each of which has the same set of neighbors as the replaced vertex. The new pair of vertices are called false twins of each other.
  3. Replace any vertex of the graph by a pair of vertices, each of which has as its neighbors the neighbors of the replaced vertex together with the other vertex of the pair. The new pair of vertices are called true twins of each other.

• They are the graphs that may be completely decomposed into cliques and stars (complete bipartite graphs K_1,q) by a split decomposition. In this decomposition, one finds a partition of the graph into two subsets, such that the edges separating the two subsets form a complete bipartite subgraph, forms two smaller graphs by replacing each of the two sides of the partition by a single vertex, and recursively partitions these two subgraphs.

• They are the graphs that have rank-width one, where the rank-width of a graph is defined as the minimum, over all hierarchical partitions of the vertices of the graph, of the maximum rank among certain submatrices of the graph's adjacency matrix determined by the partition.