Size reconstructibility of graphs

The deck of a graph $G$ is given by the multiset of (unlabelled) subgraphs $\{G-v:v\in V(G)\}$. The subgraphs $G-v$ are referred to as the cards of $G$. Brown and Fenner recently showed that, for $n\geq29$, the number of edges of a graph $G$ can be computed from any deck missing 2 cards. We show that, for sufficiently large $n$, the number of edges can be computed from any deck missing at most $\frac1{20}\sqrt{n}$ cards.Comment: 15 page

Switching Reconstruction of Digraphs

Switching about a vertex in a digraph means to reverse the direction of every edge incident with that vertex. Bondy and Mercier introduced the problem of whether a digraph can be reconstructed up to isomorphism from the multiset of isomorphism types of digraphs obtained by switching about each vertex. Since the largest known non-reconstructible oriented graphs have 8 vertices, it is natural to ask whether there are any larger non-reconstructible graphs. In this paper we continue the investigation of this question. We find that there are exactly 44 non-reconstructible oriented graphs whose underlying undirected graphs have maximum degree at most 2. We also determine the full set of switching-stable oriented graphs, which are those graphs for which all switchings return a digraph isomorphic to the original

Isomorph-free generation of 2-connected graphs with applications

Many interesting graph families contain only 2-connected graphs, which have ear decompositions. We develop a technique to generate families of unlabeled 2-connected graphs using ear augmentations and apply this technique to two problems. In the first application, we search for uniquely K_r-saturated graphs and find the list of uniquely K_4-saturated graphs on at most 12 vertices, supporting current conjectures for this problem. In the second application, we verifying the Edge Reconstruction Conjecture for all 2-connected graphs on at most 12 vertices. This technique can be easily extended to more problems concerning 2-connected graphs.Comment: 15 pages, 3 figures, 4 table

Polynomial reconstruction : old and new techniques

The Polynomial Reconstruction Problem (PRP) asks whether for a graph G of order at least three, the characteristic polynomial can be reconstructed from the p-deck PD(G) of characteristic polynomials of the one-vertex-deleted subgraphs. The problem is still open in general but has been proved for certain classes of graphs. We discuss the tools and techniques most commonly used and survey the main positive results obtained so far, pointing out the classes of graphs for which we know that the PRP has a positive resolution.peer-reviewe

On a new digraph reconstruction conjecture

Some classes of digraphs are reconstructed from the point-deleted subdigraphs for each of which the degree pair of the deleted point is also known. Several infinite families of known counterexamples to the Digraph Reconstruction Conjecture (DRC) turn out to be reconstructible in this sense. A new conjecture concerning reconstruction of digraphs in this sense is proposed and none of the known counterexample pairs to the DRC is a counterexample pair to this new conjecture

Edge reconstruction of the Ihara zeta function

We show that if a graph $G$ has average degree $\bar d \geq 4$, then the Ihara zeta function of $G$ is edge-reconstructible. We prove some general spectral properties of the edge adjacency operator $T$: it is symmetric for an indefinite form and has a "large" semi-simple part (but it can fail to be semi-simple in general). We prove that this implies that if $\bar d>4$, one can reconstruct the number of non-backtracking (closed or not) walks through a given edge, the Perron-Frobenius eigenvector of $T$ (modulo a natural symmetry), as well as the closed walks that pass through a given edge in both directions at least once. The appendix by Daniel MacDonald established the analogue for multigraphs of some basic results in reconstruction theory of simple graphs that are used in the main text.Comment: 19 pages, 2 pictures, in version 2 some minor changes and now including an appendix by Daniel McDonal