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Concept
Distance-hereditary graph
Summary
In graph theory, a branch of discrete mathematics, a distance-hereditary graph (also called a completely separable graph) is a graph in which the distances in any connected induced subgraph are the same as they are in the original graph. Thus, any induced subgraph inherits the distances of the larger graph.
Distance-hereditary graphs were named and first studied by , although an equivalent class of graphs was already shown to be perfect in 1970 by Olaru and Sachs.
It has been known for some time that the distance-hereditary graphs constitute an intersection class of graphs, but no intersection model was known until one was given by .
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 can 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 five or more has at least two 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 can be built up from a single vertex by a sequence of the following three operations, as shown in the illustration:
Add a new pendant vertex connected by a single edge to an existing vertex of the graph.
Official source
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