Reconstructing infinite graphs

Abstract

It is a well-known conjecture of S. M. Ulam that any finite graph of order at least three can be reconstructed from its maximal vertex-deleted subgraphs. Formally (writing G υ for Gv) Ulam's Conjecture states: if G and H are finite graphs of order at least three such that there is a bijection σ: V(G)-> V(H) with the property (1) G Ό s* H aM for all v e V(G) , then G = H. This conjecture has not been proved in general, although it was shown by P. J. Kelly to be true for disconnected graphs and trees and has also been verified for several other classes of graphs. The purpose of this paper is to examine Ulam's Conjecture for infinite graphs. (It is trivial to determine, from any G v , whether or not a graph G is infinite.) Results are obtained which can loosely be viewed as extensions of Kelly's work on disconnected graphs and trees. In §2 it is shown that infinite graphs G and H satisfying (1) must have the same finite components, occurring with the same multiplicity. Corollaries of this are that if G either has only finite components, or has some finitely occurring finite component, then G ~ H. In § 3 the conjecture is proved for m-coherent locally finite trees, where m is finite and greater than one. This furnishes a partial solution to the reconstruction problem for infinite trees, raised by C. St. J. A. Nash-Williams. We have used the language of reconstruction in our proofs. However, it should be noted that the results are existential in nature and not algorithmic. Throughout the paper G and H will denote infinite graphs satisfying condition (1) of Ulam's Conjecture. Any notation and terminology not defined can be found in Harary [3]. 2* Disconnected graphs* We denote by c(G) the number of components of G, and by c(G; K) the number of components of G that are isomorphic to K. A finite connected graph J is called a K-producer if c(J v ; K) > 0 for all ve V(J). (Since J has a non-cutvertex, J must be regular and of order one more than the order of K; hence K determines / up to isomorphism.) An endvertex of G is a vertex of degree one. LEMMA 2.1. If L is infinite and connected and K is finite, then there is an infinite set S £ V(L) such that c(L v ; K) = 0 for all ve S.

Reconstructing the number of blocks of an infinite graph

“…By a result of Bondy and Hemminger [5], the number of components which are isomorphic to some given finite graph is the same for G and H, and thus G and H must have the same number of isolated vertices and the same number of isolated edges. From this, together with Lemma 1, one obtains that…”

Non-reconstructible locally finite graphs

“…The singletons tvu for v P V pGq and sets of the form Cpω, Sq Y Ωpω, Sq generate a compact metrizable topology on the set V pGq Y ΩpGq, which is known in the literature as |G|. 2 This topology allows us to talk about the closure of a set of vertices X Ă V pGq, denoted by X. Write BpXq " XzX " X X ΩpXq for the boundary of X: the collection of all ends in the closure of X.…”

Understanding chicken walks on n × n grid: Hamiltonian paths, discrete dynamics, and rectifiable paths

“…j 0 1//, i.e., K.1, . j 0 1//; (c 14 ) take .2n j 0 C 1/ steps in the direction 1j 0 d 1. j 0 1/ to reach the last column; (c 15 ) take .2n 1/ steps in the direction 2.2n/ d 1.2n/ to reach the last row; (c 16 ) take one step in the direction .2n/.2n 1/ d .2n/.2n/ ; (c 17 ) take .2n 2/ steps in the direction .2n 1/.2n 1/ d .2n/.2n 1/ ; (c 18 ) take one step in the direction 2.2n 2/ d 2.2n 1/ ; (c 19 ) take .2n 2/ steps in the direction 3.2n 2/ d 2.2n 2/ ; (c 20 ) repeat the steps similar to the steps (c 16 ) to (c 19 ) such that the chicken is located in the cell ..2n/, . j 0 C 3//, i.e., K..2n/, .…”

“…2n 3/ ; (b16 ) repeat the steps similar to the steps (b 7 ) to (b15 ) to reach the cell ..2n i 0 C 1/, 1/; (b 17 ) continue for i 0 steps in the same direction to reach the cell .1, 1/; (b 18 ) take .2n 1/ steps in the direction12 d 11 to reach the last column; (b 19 ) take one step in the direction 2.2n/ d 1.2n/ ; (b 20 ) take .2n 2/ steps in the direction 22 d 2.2n/ ; (b 21 ) take one step in the direction 32 d 22 ; (b 22 ) take .2n 2/ steps in the direction 33 d 32 to reach the last column; (b 23 ) repeat the steps similar to the steps (b 19 ) to (b 22 ) such that the chicken is located in the cell ..i 0 3/, 2n/; (b 24 ) take two steps in the direction .i 0 2/.2n/ d .i 0 3/.2n/ ; (b 25 ) take one step in the direction .i 0 1/.2n 1/ d .i 0 1/.2n/ ; (b 26 ) take one step in the direction .i 0 2/.2n 1/ d .i 0 1/.2n 1/ ; (b 27 ) take one step in the direction .i 0 2/.2n 2/ d .i 0 2/.2n 1/ ; (b 28 ) take one step in the direction .i 0 1/.2n 2/ d .i 0 2/.2n 2/ ; (b 29 ) repeat the steps similar to the steps (b 25 ) to (b 28 ) to reach the cell ..i 0 1/, 2/, i.e., K ..i 0 1/, 2/; (b 30 ) take one step in the direction i 0 2 d .i 0 1/2 ; and (b 31 ) take . j 0 1/ steps in the direction i 0 3 d i 0 ,2 to reach the maximum distance configuration.…”