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The Graph Minors Structure Theorem of Robertson and Seymour asserts that, for every graph $H,$ every $H$-minor-free graph can be obtained by clique-sums of ``almost embeddable'' graphs. Here a graph is ``almost embeddable'' if it can be…
In this paper, we investigate Hamiltonian path problem in the context of split graphs, and produce a dichotomy result on the complexity of the problem. Our main result is a deep investigation of the structure of $K_{1,4}$-free split graphs…
We show that the independence number of a countably infinite HH-homogeneous graph that does not contain the Rado graph as a spanning subgraph is finite and present a classification of MB-homogeneous graphs up to bimorphism-equivalence as a…
The class of all even-hole-free graphs has unbounded tree-width, as it contains all complete graphs. Recently, a class of (even-hole, $K_4$)-free graphs was constructed, that still has unbounded tree-width [Sintiari and Trotignon, 2019].…
We study the cohomology of the hairy graph complexes which compute the rational homotopy of embedding spaces, generalizing the Vassiliev invariants of knot theory. We provide spectral sequences converging to zero whose first pages contain…
A graph is $(H_1,H_2)$-free for a pair of graphs $H_1,H_2$ if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. In 2001, Kr\'al', Kratochv\'{\i}l, Tuza, and Woeginger initiated a study into the complexity of Colouring for…
$Hom(G,H)$ is a polyhedral complex defined for any two undirected graphs $G$ and $H$. This construction was introduced by Lov\'asz to give lower bounds for chromatic numbers of graphs. In this paper we initiate the study of the topological…
For a graph $H$, its homomorphism density in graphs naturally extends to the space of two-variable symmetric functions $W$ in $L^p$, $p\geq e(H)$, denoted by $t(H,W)$. One may then define corresponding functionals…
We introduce the tree distance, a new distance measure on graphs. The tree distance can be computed in polynomial time with standard methods from convex optimization. It is based on the notion of fractional isomorphism, a characterization…
A graph is $H$-free if it does not contain an induced subgraph isomorphic to $H$. We denote by $P_k$ and $C_k$ the path and the cycle on $k$ vertices, respectively. In this paper, we prove that 4-COLORING is NP-complete for $P_7$-free…
An NP-complete coloring or homomorphism problem may become polynomial time solvable when restricted to graphs with degrees bounded by a small number, but remain NP-complete if the bound is higher. For instance, 3-colorability of graphs with…
This paper introduces the notion of involution module, the first generalization of the modular decomposition of 2-structure which has a unique linear-sized decomposition tree. We derive an O(n^2) decomposition algorithm and we take…
We construct a pair of non-isomorphic, bipartite graphs which are not distinguished by counting the number of homomorphisms to any tree. This answers a question motivated by Atserias et al. (LICS 2021). In order to establish the…
Recently there has been growing interest in discrete homotopies and homotopies of graphs beyond treating graphs as 1-dimensional simplicial spaces. One such type of homotopy is $\times$-homotopy. Recent work by Chih-Scull has developed a…
Let $G$ be a graph that contains an induced subgraph $H$. A retraction from $G$ to $H$ is a homomorphism from $G$ to $H$ that is the identity function on $H$. Retractions are very well-studied: Given $H$, the complexity of deciding whether…
A homomorphism from a graph G to a graph H is locally bijective, surjective, or injective if its restriction to the neighborhood of every vertex of G is bijective, surjective, or injective, respectively. We prove that the problems of…
Recently, the theory of dense graph limits has received attention from multiple disciplines including graph theory, computer science, statistical physics, probability, statistics, and group theory. In this paper we initiate the study of the…
Let $M$ be a left $R$-module. We define the \emph{homomorphism submodule graph} $\Gamma_{\mathrm{Hom}}(M)$ as the simple graph whose vertices are the proper submodules of $M$, with an edge between distinct vertices $N_1$ and $N_2$ if and…
We develop a limit theory for $1$-cochains of complete graphs with coefficients from a finite abelian group. We prove an analogue of the large deviation principle of Chatterjee and Varadhan for random cochains. We use these new tools to…
A family T of digraphs is a complete set of obstructions for a digraph H if for an arbitrary digraph G the existence of a homomorphism from G to H is equivalent to the non-existence of a homomorphism from any member of T to G. A digraph H…