A note on Steiner intervals and betweenness

Manoj Changat (Author), Anandavally K. Lakshmikuttyamma (Author), Joseph Mathews (Author), Iztok Peterin (Author), Prasanth G. Narasimha-Shenoi (Author), Aleksandra Tepeh (Author)

Abstract

Geodetka in geodetski interval, ki je sestavljen iz vseh vozlišč, ki pripadajo kakšni geodetki med fiksnim parom vozlišč v povezanem grafu ▫$G$▫, sta sestavni del metrične teorije grafov. Prav tako je znano, da je Steinerjevo drevo (multi) množice s ▫$k$▫ (▫$k>2$▫) vozlišči, posplošitev geodetke. V (B. Brešar, M. Changat, J. Mathews, I. Peterin, P. G. Narasimha-Shenoi, A. Tepeh Horvat, Steiner intervals, geodesic intervals, and betweenness, Discrete Math. 309 (2009) 6114--6125) so se avtorji ukvarjali s ▫$k$▫-Steinerjevimi intervali ▫$S(u_{1},u_{2},\ldots, u_{k})$▫ povezanih grafov (▫$k \geq 3$▫) kot ▫$k$▫-arnimi posplošitvami geodetskih intervalov. Analogno sta bila iz binarne na ▫$k$▫-arno funkcijo posplošena tudi vmesnostni aksiom (b2) in monotoni aksiom(m) kot: za vsa vozlišča ▫$u_{1}, \ldots, u_{k}, x, x_{1}, \ldots, x_{k} \in V(G)$▫, ki niso nujno različna ▫$$(b2)\quad x \in S(u_{1}, u_{2}, \ldots, u_{k}) \Rightarrow S(x, u_{2}, \ldots, u_{k}) \subseteq S(u_{1}, u_{2}, \ldots, u_{k}),$$▫ ▫$$(m) \quad x_{1}, \ldots, x_{k} \in S(u_{1}, \ldots, u_{k})\Rightarrow S(x_{1}, \ldots,x_{k}) \subseteq S(u_{1}, \ldots, u_{k}).$$▫ Avtorji so v zgoraj omenjenem članku domnevali, da ▫$3$▫-Steinerjev interval povezanega grafa ▫$G$▫ zadošča vmesnostnemu aksiomu (b2) natanko tedaj, ko je vsak blok grafa ▫$G$▫ geodetski z diametrom največ 2. V tem delu dokažemo to domnevo. Pri tem dodatno dokažemo, da v vsakem geodetskem bloku z diametrom vsaj 3 obstaja izometrični cikel dolžine ▫$2k+1$▫, ▫$k>2$▫. Prav tako predstavimo dodaten aksiom (b2(2)), ki je smiseln le za 3-Steinerjeve intervale in pokažemo, da je le ta ekvivalenten monotonemu aksiomu.

Keywords

matematika;teorija grafov;Steinerjev interval;geodetski graf;vmesnost;mathematics;graph theory;Steiner interval;geodetic graph;betweenness;

Data

Language:	English
Year of publishing:	2011
Typology:	1.01 - Original Scientific Article
Organization:	UM FERI - Faculty of Electrical Engineering and Computer Science
UDC:	519.17
COBISS:	16078937
ISSN:	0012-365X
Views:	47
Downloads:	30
Average score:	0 (0 votes)
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Other data

Secondary language:	Slovenian
Secondary title:	Notica o 3-Steinerjevih intervalih in vmesnost
Secondary abstract:	The geodesic and geodesic interval, namely the set of all vertices lying on geodesics between a pair of vertices in a connected graph, is a part of folklore in metric graph theory. It is also known that Steiner tree of a (multi) set with ▫$k$▫ (▫$k>2$▫) vertices, generalizes geodesics. In (B. Brešar, M. Changat, J. Mathews, I. Peterin, P. G. Narasimha-Shenoi, A. Tepeh Horvat, Steiner intervals, geodesic intervals, and betweenness, Discrete Math. 309 (2009) 6114-6125) the authors studied the ▫$k$▫-Steiner intervals ▫$S(u_{1}, u_{2}, \ldots, u_{k})$▫ on connected graphs (▫$k \geq 3$▫) asthe ▫$k$▫-ary generalization of the geodesic intervals. The analogous betweenness axiom (b2) and the monotone axiom (m) were generalized from binary to ▫$k$▫-ary functions as: for any ▫$u_{1}, \ldots, u_{k}, x, x_{1}, \ldots, x_{k} \in V(G)$▫ which are not necessarily distinct, ▫$$(b2) \quad x \in S(u_{1}, u_{2}, \ldots, u_{k}) \Rightarrow S(x,u_{2}, \ldots, u_{k}) \subseteq S(u_{1}, u_{2}, \ldots, u_{k}),$$▫ ▫$$(m) \quad x_{1}, \ldots, x_{k} \in S(u_{1}, \ldots, u_{k}) \Rightarrow S(x_{1}, \ldots, x_{k}) \subseteq S(u_{1}, \ldots, u_{k}). ▫$$ The authors conjectured that the ▫$3$▫-Steiner interval on a connected graph ▫$G$▫ satisfies the betweenness axiom (b2) if and only if each block of ▫$G$▫ is geodetic of diameter at most 2. In this paper we settle this conjecture. For this we show that there exists an isometric cycle of length ▫$2k+1$▫, ▫$k>2$▫, in every geodetic block of diameter at least 3. We also introduce another axiom (b2(2)), which is meaningful only to ▫$3$▫-Steiner intervals and show that this axiom is equivalent to the monotone axiom.
Secondary keywords:	matematika;teorija grafov;Steinerjev interval;geodetski graf;vmesnost;
URN:	URN:SI:UM:
Type (COBISS):	Not categorized
Pages:	Str. 2601-2609
Volume:	Vol. 311
Issue:	iss. 22
Chronology:	2011
ID:	1475889