On graphs with the same restricted U-polynomial and the U-polynomial for rooted graphs

In this abstract, we construct explicitly, for every k, pairs of non-isomorphic trees with the same restricted U-polynomial; by this we mean that the polynomials agree on terms with degree at most k. The construction is done purely in algebraic terms, after introducing and studying a generalization of the U-polynomial to rooted graphs.Peer ReviewedPostprint (author's final draft

Reconstructing trees from small cards

The $\ell$-deck of a graph $G$ is the multiset of all induced subgraphs of $G$ on $\ell$ vertices. In 1976, Giles proved that any tree on $n\geq 6$ vertices can be reconstructed from its $\ell$-deck for $\ell \geq n-2$. Our main theorem states that it is enough to have $\ell\geq (8/9+o(1))n$, making substantial progress towards a conjecture of N\'ydl from 1990. In addition, we can recognise connectedness from the $\ell$-deck if $\ell\geq 9n/10$, and reconstruct the degree sequence from the $\ell$-deck if $\ell\ge \sqrt{2n\log(2n)}$. All of these results are significant improvements on previous bounds.Comment: 24 pages, fixed several typo

Homomorphism Tensors and Linear Equations

Lov\'asz (1967) showed that two graphs $G$ and $H$ are isomorphic if and only if they are homomorphism indistinguishable over the class of all graphs, i.e. for every graph $F$, the number of homomorphisms from $F$ to $G$ equals the number of homomorphisms from $F$ to $H$. Recently, homomorphism indistinguishability over restricted classes of graphs such as bounded treewidth, bounded treedepth and planar graphs, has emerged as a surprisingly powerful framework for capturing diverse equivalence relations on graphs arising from logical equivalence and algebraic equation systems. In this paper, we provide a unified algebraic framework for such results by examining the linear-algebraic and representation-theoretic structure of tensors counting homomorphisms from labelled graphs. The existence of certain linear transformations between such homomorphism tensor subspaces can be interpreted both as homomorphism indistinguishability over a graph class and as feasibility of an equational system. Following this framework, we obtain characterisations of homomorphism indistinguishability over two natural graph classes, namely trees of bounded degree and graphs of bounded pathwidth, answering a question of Dell et al. (2018).Comment: 33 pages, accepted for ICALP 202