Fault-diameter of Cartesian product of graphs and Cartesian graph bundles

Iztok Banič (Author), Janez Žerovnik (Author)

Abstract

Cartesian graph bundles is a class of graphs that is a generalization of the Cartesian graph products. Let ▫$G$▫ be a ▫$k_G$▫-connected graph and ▫${\mathcal{D}}_c(G)$▫ denote the diameter of ▫$G$▫ after deleting any of its ▫$c < k_G$▫ vertices. We prove that if ▫$G_1, G_2, \dots, G_q$▫ are ▫$k_1$▫-connected, ▫$k_2$▫-connected,...,▫$k_q$▫-connected graphs and ▫$0 \leq a_1 < k_1$▫, ▫$0 \leq a_2 < k_2$▫,...,▫$0 \leq a_q < k_q$▫ and ▫$a = a_1 + a_2 + \dots + a_q + (q-1)$▫, then the fault diameter of ▫$G$▫, a Cartesian product of ▫$G_1$▫, ▫$G_2$▫,...,▫$G_q$▫, with ▫$a$▫ faulty nodes is ▫${\mathcal{D}}_{a}(G) \leq {\mathcal{D}}_{a_1}(G_1)+{\mathcal{D}}_{a_2}(G_2) + \dots + {\mathcal{D}}_{a_q}(G_q) + 1$▫. We also show that ▫${\mathcal{D}}_{a+b+1}(G) \leq {\mathcal{D}}_a(F) + {\mathcal{D}}_b(B) + 1$▫ if ▫$G$▫ is a graph bundle with fibre ▫$F$▫ over base ▫$B$▫, ▫$a \leq k_F$▫, and ▫$b \leq k_B$▫. As an auxiliary result we prove that connectivity of graph bundle ▫$G$▫ is at least ▫$k_F+k_B$▫.

Keywords

mathematics;graph theory;Cartesian graph bundles;Cartesian graph products;fault diameter;interconnection network;

Data

Language:	English
Year of publishing:	2006
Typology:	0 - Not set
Organization:	UM FS - Faculty of Mechanical Engineering
UDC:	519.17
COBISS:	14564441
ISSN:	1318-4865
Views:	662
Downloads:	24
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	Unknown
URN:	URN:SI:UM:
Type (COBISS):	Not categorized
Pages:	str. 1-17
Volume:	ǂVol. ǂ44
Issue:	ǂšt. ǂ1016
Chronology:	2006
ID:	67265