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

Universal targets for homomorphisms of edge-colored graphs

A $k$-edge-colored graph is a finite, simple graph with edges labeled by numbers $1,\ldots,k$. A function from the vertex set of one $k$-edge-colored graph to another is a homomorphism if the endpoints of any edge are mapped to two different vertices connected by an edge of the same color. Given a class $\mathcal{F}$ of graphs, a $k$-edge-colored graph $\mathbb{H}$ (not necessarily with the underlying graph in $\mathcal{F}$) is $k$-universal for $\mathcal{F}$ when any $k$-edge-colored graph with the underlying graph in $\mathcal{F}$ admits a homomorphism to $\mathbb{H}$. We characterize graph classes that admit $k$-universal graphs. For such classes, we establish asymptotically almost tight bounds on the size of the smallest universal graph. For a nonempty graph $G$, the density of $G$ is the maximum ratio of the number of edges to the number of vertices ranging over all nonempty subgraphs of $G$. For a nonempty class $\mathcal{F}$ of graphs, $D(\mathcal{F})$ denotes the density of $\mathcal{F}$, that is the supremum of densities of graphs in $\mathcal{F}$. The main results are the following. The class $\mathcal{F}$ admits $k$-universal graphs for $k\geq2$ if and only if there is an absolute constant that bounds the acyclic chromatic number of any graph in $\mathcal{F}$. For any such class, there exists a constant $c$, such that for any $k \geq 2$, the size of the smallest $k$-universal graph is between $k^{D(\mathcal{F})}$ and $ck^{\lceil D(\mathcal{F})\rceil}$. A connection between the acyclic coloring and the existence of universal graphs was first observed by Alon and Marshall (Journal of Algebraic Combinatorics, 8(1):5-13, 1998). One of their results is that for planar graphs, the size of the smallest $k$-universal graph is between $k^3+3$ and $5k^4$. Our results yield that there exists a constant $c$ such that for all $k$, this size is bounded from above by $ck^3$

Classes of Intersection Digraphs with Good Algorithmic Properties

While intersection graphs play a central role in the algorithmic analysis of hard problems on undirected graphs, the role of intersection digraphs in algorithms is much less understood. We present several contributions towards a better understanding of the algorithmic treatment of intersection digraphs. First, we introduce natural classes of intersection digraphs that generalize several classes studied in the literature. Second, we define the directed locally checkable vertex (DLCV) problems, which capture many well-studied problems on digraphs such as (Independent) Dominating Set, Kernel, and H-Homomorphism. Third, we give a new width measure of digraphs, bi-mim-width, and show that the DLCV problems are polynomial-time solvable when we are provided a decomposition of small bi-mim-width. Fourth, we show that several classes of intersection digraphs have bounded bi-mim-width, implying that we can solve all DLCV problems on these classes in polynomial time given an intersection representation of the input digraph. We identify reflexivity as a useful condition to obtain intersection digraph classes of bounded bi-mim-width, and therefore to obtain positive algorithmic results

On infinite-finite duality pairs of directed graphs

The (A,D) duality pairs play crucial role in the theory of general relational structures and in the Constraint Satisfaction Problem. The case where both classes are finite is fully characterized. The case when both side are infinite seems to be very complex. It is also known that no finite-infinite duality pair is possible if we make the additional restriction that both classes are antichains. In this paper (which is the first one of a series) we start the detailed study of the infinite-finite case. Here we concentrate on directed graphs. We prove some elementary properties of the infinite-finite duality pairs, including lower and upper bounds on the size of D, and show that the elements of A must be equivalent to forests if A is an antichain. Then we construct instructive examples, where the elements of A are paths or trees. Note that the existence of infinite-finite antichain dualities was not previously known