# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
The majority strategy on graphs. (English) Zbl 0888.05025
An interval $I(u,v)$ in a graph $G$, where $u$, $v$ are vertices, is the set of all vertices $w$ of $G$ for which the equality $d(u, w)+ d(w,v)= d(u,v)$ holds, where $d$ denotes the distance. If for any three vertices $u$, $v$, $w$ of $G$ the intersection $I(u,v)\cap I(v, w)\cap I(w,u)$ consists of one vertex only, then $G$ is called a median graph. A profile of length $p$ on a graph $G$ is a finite sequence $v_1,v_2,\dots, v_p$ of vertices of $G$; its median is a vertex $x$ of $G$ for which $\sum^p_{i=1} d(x, v_i)$ is minimal. The set of all medians of a profile $\pi$ is the median set $M(\pi)$. In the paper a strategy for finding $M(\pi)$ for a given profile $\pi$ is described; it is called the majority strategy. It begins in a vertex of $G$ and consists of certain successive moves from one vertex to another along an edge. It is proved that the majority strategy produces the median set $M(\pi)$ for each profile $\pi$ of $G$, independently of the initial position, if and only if $G$ is a median graph.

##### MSC:
 05C12 Distance in graphs 05C75 Structural characterization of families of graphs 05C99 Graph theory 90B80 Discrete location and assignment
##### Keywords:
interval; median graph; profile; median set; majority strategy
Full Text:
##### References:
 [1] Avann, S. P.: Ternary distributive semi-lattices. Bull. amer. Math. soc. 54, 79 (1948) · Zbl 0099.02201 [2] Avann, S. P.: Metric ternary distributive semi-lattices. Proc. amer. Math. soc. 12, 407-414 (1961) · Zbl 0099.02201 [3] Bandelt, H. -J.; Barthélemy, J. -P.: Medians in median graphs. Discrete appl. Math. 8, 131-142 (1984) · Zbl 0536.05057 [4] Chung, F. R. K.; Graham, R. L.; Saks, M. E.: Dynamic search in graphs. Discrete algorithms and complexity, 351-387 (1987) [5] Jordan, C.: Sur LES assemblages de lignes. J. reine angew. Math. 70, 185-190 (1869) [6] Goldman, A. J.: Optimal center location in simple networks. Transport. sci. 5, 212-221 (1979) [7] Hagauer, J.; Imrich, W.; Klavžar, S.: Recognizing median graphs. Univ. of Ljubljana, IMFM 31 (1993) · Zbl 0913.68152 [8] Imrich, W.; Klavžar, S.: On the complexity of recognizing Hamming graphs and related classes of graphs. Eur. J. Combin. 17, 209-221 (1996) · Zbl 0846.05082 [9] Jha, P. K.; Slutzky, G.: Convex-expansion algorithms for recognizing and isometric embedding of median graphs. Ars combin. 34, 75-92 (1992) [10] S. Klavzar, H.M. Mulder, Median graphs: characterizations, location theory and related structures. J. Combin. Math. Combin. Computing, to appear. · Zbl 0916.68106 [11] Mcmorris, F. R.; Mulder, H. M.; Roberts, F. S.: The median procedure on median graphs. Report 9535/B (1995) · Zbl 0906.05023 [12] Mulder, H. M.: The structure of median graphs. Discrete math. 24, 197-204 (1978) · Zbl 0394.05038 [13] Mulder, H. M.: The interval function of a graph, mathematical centre tracts 132. (1980) · Zbl 0446.05039 [14] Mulder, H. M.: The expansion procedure for graphs. Contemporary methods in graph theory, 459-477 (1990) · Zbl 0744.05064 [15] Mulder, H. M.; Schrijver, A.: Median graphs and Helly hypergraphs. Discrete math. 25, 41-50 (1979) · Zbl 0395.05058 [16] Nebeský, L.: Median graphs. Comment. math. Univ. carolinae 12, 317-325 (1971) · Zbl 0215.34001 [17] Zelinka, B.: Medians and peripherians of trees. Arch. math. (Brno) 4, 87-95 (1968) · Zbl 0206.26105