Iztok Banič

Fault-diameter of Cartesian graph bundles

Cartesian graph bundles is a class of graphs that is a generalization of the Cartesian graph products. Let G be a kG-connected graph and Dc(G) denote the diameter of G after deleting any of its c < kG vertices. We prove that Da+b+1(G) ≤ Da(F) + Db(B) + 1 if G is a graph bundle with fibre F over base B, a < kF, and b < kB.

Edge fault-diameter of Cartesian product of graphs

Let G be a kG-edge connected graph and Dc(G) denote the diameter of G after deleting any of its c < kG edges. We prove that if G1, G2, . . . , Gq are k1-edge connected, k2-edge connected,. . . , kq-edge connected graphs and 0 ≤ a1 < k1, 0 ≤ a2 < k2,. . . , 0 ≤ aq < kq and a = a1 + a2 + . . . + aq + (q - 1), then the edge fault-diameter of G, the Cartesian product of G1, G2, . . . , Gq, with a faulty edges is Da(G) ≤ Da1(G1) + Da2(G2) + . . . + Daq(Gq) + 1.

Inverse limits as limits with respect to the Hausdorff metric

We show that the inverse limit of any inverse sequence of compact metric spaces and surjective bonding maps is in fact the limit of a sequence of homeomorphic copies of the same spaces with respect to the Hausdorff metric.

INVERSE LIMITS IN THE CATEGORY OF COMPACT HAUSDORFF SPACES AND UPPER SEMICONTINUOUS FUNCTIONS

We investigate inverse limits in the category \(\mathcal{CHU}\) of compact Hausdorff spaces with upper semicontinuous functions. We introduce the notion of weak inverse limits in this category and show that the inverse limits with upper semicontinuous set-valued bonding functions (as they were defined by Ingram and Mahavier [‘Inverse limits of upper semi-continuous set valued functions’, Houston J. Math. 32 (2006), 119–130]) together with the projections are not necessarily inverse limits in \(\mathcal{CHU}\) but they are always weak inverse limits in this category. This is a realisation of our categorical approach to solving a problem stated by Ingram [An Introduction to Inverse Limits with Set-Valued Functions (Springer, New York, 2012)]. 10.1017/S0004972713000245

Fault-diameter of generalized Cartesian products

Cartesian graph bundles is a class of graphs that is a generalization of the Cartesian graph products. Let G be a kG-connected graph and D_c(G) denote the diameter of G after deleting any of its c \lt kG vertices. For a product of three factors G_1, G_2 and G_3, we prove that D_a+b+c+2(G) \lt D_a(G_1) + D_b(G_2) + D_c(G_3) + 1. We indicate how analogous proof gives the upper bound D_a+b+1(G) \lt D_a(G_1) + D_b(G_2) + 1 for the product of two factors. Finally, we show that D_a+b+1(G) \lt D_a(F) + D_b(B)+1 if G is a graph bundle with fibre F over base B, a \lt k_F,and b \lt k_B.