Steiner intervals, geodesic intervals, and betweenness

Boštjan Brešar (Author), Manoj Changat (Author), Joseph Mathews (Author), Iztok Peterin (Author), Prasanth G. Narasimha-Shenoi (Author), Aleksandra Tepeh (Author)

Abstract

Koncept ▫$k$▫-Steinerjevih intervalov naravno posplošuje geodetske (binarne) intervale. Definiran je kot preslikava ▫$S: V\times \cdots \times V \longrightarrow 2^V$▫, kjer je ▫$S(u_1, \dots ,u_k)$▫ množica tistih vozlišč grafa ▫$G$▫, ki ležijo na kakem Steinerjevem drevesu glede na multimnožico ▫$W = \{u_1, \dots ,u_k\}$▫ vozlišč iz ▫$G$▫. V tem članku za vsako naravno število ▫$k$▫ dokažemo karakterizacijo razreda tistih grafov, v katerih imajo vsi ▫$k$▫-Steinerjevi intervali t.i. lastnost unije, ki pravi, da ▫$S(u_1,\ldots, u_k)$▫ sovpada z unijo geodetskih intervalov ▫$I(u_i,u_j)$▫ med vsemi pari vozlišč iz ▫$W$▫. Izkaže se, da tedaj, ko je ▫$k>3$▫, ta razred sovpada z razredom grafov, v katerih ▫$k$▫-Steinerjev interval zadošča aksiomu monotonosti(m), kot tudi z razredom grafov, v katerih ▫$k$▫-Steinerjev interval zadošča aksiomu (b2), ki sta pogoja iz teorije vmesnosti. In sicer preslikava ▫$S$▫ zadošča aksiomu (m), če iz ▫$x_1, \dots ,x_k \in S(u_1, \dots ,u_k)$▫ sledi ▫$S(x_1, \dots ,x_k) \subseteq S(u_1, \dots ,u_k)$▫; ter ▫$S$▫ zadošča (b2), če iz ▫$x \in S(u_1,u_2, \dots ,u_k)$▫ sledi ▫$S(x,u_2, \dots ,u_k) \subseteq S(u_1, \dots ,u_k)$▫. V primeru ▫$k=3$▫ so ti trije razredi grafov različni in za razreda grafov, v katerih Steinerjev interval zadošča lastnosti unije oz. aksiomu monotonosti (m), dokažemo strukturni karakterizaciji. Prav tako predstavimo več delnih ugotovitev za razred grafov, v katerih 3-Steinerjev interval zadošča aksimu (b2), ki vodijo do domneve, da so to natanko tisti grafi, v katerih je vsak blok geodetski graf z diametrom 2.

Keywords

matematika;teorija grafov;Steinerjev interval;geodetski interval;razdalja;vmesnost;monotonost;bločni graf;mathematics;graph theory;Steiner interval;geodesic interval;distance;betweenness;monotonicity;block graph;

Data

Language:	English
Year of publishing:	2009
Typology:	1.01 - Original Scientific Article
Organization:	UM FERI - Faculty of Electrical Engineering and Computer Science
UDC:	519.17
COBISS:	15334233
ISSN:	0012-365X
Views:	64
Downloads:	30
Average score:	0 (0 votes)
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Other data

Secondary language:	Slovenian
Secondary title:	Steinerjevi intervali, geodetski intervali in vmesnost
Secondary abstract:	The concept of the ▫$k$▫-Steiner interval is a natural generalization of the geodesic (binary) interval. It is defined as a mapping ▫$S: V \times \cdots \times V \longrightarrow 2^V$▫ such that ▫$S(u_1,...,u_k)$▫ consists of all vertices in ▫$G$▫ that lie on some Steiner tree with respect to a multiset ▫$W =\{u_1,...,u_k\}$▫ of vertices from ▫$G$▫. In this paper we obtain, for each ▫$k$▫, a characterization of the class of graphs in which every ▫$k$▫-Steiner interval ▫$S$▫ has the so-called union property, which says that ▫$S(u_1,...,u_k)$▫ coincides with the union of geodesic intervals ▫$I(u_i,u_j)$▫ between all pairs from ▫$W$▫. It turns out that, as soon as ▫$k>3$▫, this class coincides with the class of graphs in which the ▫$k$▫-Steiner interval enjoys the monotone axiom (m), respectively (b2) axiom, the conditions from betweenness theory. Notably, ▫$S$▫ satisfies (m), if ▫$x_1,...,x_k \in S(u_1,...,u_k)$▫ implies ▫$S(x_1,...,x_k) \subseteq S(u_1,...,u_k)$▫, and ▫$S$▫ satisfies (b2) if ▫$x \in S(u_1,u_2,...,u_k)$▫ implies ▫$S(x,u_2,...,u_k) \subseteq S(u_1,...,u_k)$▫. In the case ▫$k=3$▫, these three classes are different, and we give structural characterizations of graphs for which their Steiner interval ▫$S$▫ satisfies the union property as well as the monotone axiom (m). We also prove several partial observations on the class ofgraphs in which the 3-Steiner interval satisfies (b2), which lead to the conjecture that these are precisely the graphs in which every block is a geodetic graph with diameter at most two.
Secondary keywords:	matematika;teorija grafov;Steinerjev interval;geodetski interval;razdalja;vmesnost;monotonost;bločni graf;
URN:	URN:SI:UM:
Type (COBISS):	Not categorized
Pages:	str. 6114-6125
Volume:	ǂVol. ǂ309
Issue:	ǂiss. ǂ20
Chronology:	2009
ID:	1474523