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Distance-hereditary graphs. (English) Zbl 0605.05024
A distance-hereditary graph is a connected graph in which every induced path is isometric, that is, the distance between any two vertices in an induced path equals their distance in the graph. A block graph is a conncted graph in which every block (i.e., maximal 2-connected subgraph) is complete. Let G be a graph with distance function d. Then d is said to satisfy the four-point condition if for any four vertices u, v, w, x the larger two of the three distance sums $$d(u,v)+d(w,x)$$, $$d(u,w)+d(v,x)$$ and $$d(u,x)+d(v,w)$$ are equal. An induced subgraph H of a graph G is said to be an isometric subgraph if for any two vertices u, v of H the distance between u and v in H equals the distance between u and v in G. The authors show that if G is a connected graph having distance function d, then the following conditions are equivalent: (i) G is a block graph, (ii) d satisfies the four-point condition and (iii) neither $$K_ 4$$ minus an edge nor any circuit $$C_ n$$ with $$n\geq 4$$ is an isometric subgraph of G. It is also shown that the finite distance-hereditary graphs of order at least 2 are precisely those graphs which are obtained by applying a sequence of extensions (which are described in the paper) to $$K_ 2$$. Several consequences of this result are addressed. Moreover, distance-hereditary graphs are characterized in terms of the distance function d, or via forbidden isometric subgraphs.
Reviewer: O.Oellermann

##### MSC:
 05C38 Paths and cycles 05C75 Structural characterization of families of graphs
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##### References:
 [1] \scH. J. Bandelt, Characterizing median graphs, European J. Math., in press. · Zbl 0536.05057 [2] Blumenthal, L.M, () [3] Buneman, P, A note on the metric properties of trees, J. combin. theory ser. B, 17, 48-50, (1974) · Zbl 0286.05102 [4] Burlet, M; Uhry, J.P, Parity graphs, Ann. discrete math., 16, 1-26, (1982) · Zbl 0496.05044 [5] Corneil, D.G; Lerchs, H; Burlingham, L.Stewart, Complement reducible graphs, Discrete appl. math., 3, 163-174, (1981) · Zbl 0463.05057 [6] Howorka, E, A characterization of distance-hereditary graphs, Quart. J. math. Oxford ser. 2, 28, 417-420, (1977) · Zbl 0376.05040 [7] Howorka, E, A characterization of ptolemaic graphs; survey of results, (), 355-361 [8] Howorka, E, On metric properties of certain clique graphs, J. combin theory ser. B, 27, 67-74, (1979) · Zbl 0337.05138 [9] Howorka, E, A characterization of ptolemaic graphs, J. graph theory, 5, 323-331, (1981) · Zbl 0437.05046 [10] Jamison-Waldner, R.E, Convexity and block graphs, Congressus numerantium, 33, 129-142, (1981) · Zbl 0495.05056 [11] Jamison-Waldner, R.E, A perspective on abstract convexity: classifying alignments by varieties, (), 113-150 · Zbl 0482.52001 [12] Jung, H.A, On a class of posets and the corresponding comparability graphs, J. combin. theory ser. B, 24, 125-133, (1978) · Zbl 0382.05045 [13] Kay, D.C; Chartrand, G, A characterization of certain ptolemaic graphs, Canad. J. math., 17, 342-346, (1965) · Zbl 0139.17301 [14] Melter, R.A; Tomescu, I, Isometric embeddability for graphs, Ars combin., 12, 111-115, (1981) · Zbl 0492.05026 [15] Mulder, H.M, (), Math. Centre Tracts 132 [16] Simões Pereira, J.M.S, A note on the tree realizability of a distance matrix, J. combin. theory, 6, 303-310, (1969) · Zbl 0177.26903 [17] Soltan, V.P, d-convexity in graphs, Soviet math. dokl., 28, 419-421, (1983) · Zbl 0553.05060 [18] Soltan, V.P, (), [Russian] [19] Sumner, D.P, Dacey graphs, J. austral. math. soc., 18, 4, 492-502, (1974) · Zbl 0314.05108
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