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The Minimum Path Cover problem on directed acyclic graphs (DAGs) is a classical problem that provides a clear and simple mathematical formulation for several applications in different areas and that has an efficient algorithmic solution. In…

For simply laced $SU(3)$ graphs we offer a geometric understanding of the path creation and annihilation operators for $SU(3)$ in terms of creation and annihilation of sequences of three vertices forming triangular cells or collapsed…

For a given finite graph $G$ of minimum degree at least $k$, let $G_{p}$ be a random subgraph of $G$ obtained by taking each edge independently with probability $p$. We prove that (i) if $p \ge \omega/k$ for a function $\omega=\omega(k)$…

Given a collection $\mathcal{G} =\{G_1,G_2,\dots,G_m\}$ of graphs on the common vertex set $V$ of size $n$, an $m$-edge graph $H$ on the same vertex set $V$ is transversal in $\mathcal{G}$ if there exists a bijection $\varphi…

Let $G$ be a graph of order $n$. A path decomposition $\mathcal{P}$ of $G$ is a collection of edge-disjoint paths that covers all the edges of $G$. Let $p(G)$ denote the minimum number of paths needed in a path decomposition of $G$. Gallai…

Let P be a closed triangulated manifold, dim P=n. We consider the group of simplicial 1-chains C_1(P) and the homology group H_1(P). We also use some nonnegative weighting function L on C_1(P). For any homological class from H_1(P) method…

We show that an algebra over a cyclic operad supplied with an additional linear algebra datum called Hodge decomposition admits a minimal model whose structure maps are given in terms of summation over trees. This minimal model is unique up…

We prove that the number of Hamiltonian paths on the complement of an acyclic digraph is equal to the number of cycle covers. As an application, we obtain a new expansion of the chromatic symmetric function of incomparability graphs in…

In this article, we show that magnitude homology and path homology are closely related, and we give some applications. We define differentials ${\mathrm MH}^{\ell}_k(G) \longrightarrow {\mathrm MH}^{\ell-1}_{k-1}(G)$ between magnitude…

We construct explicit minimal models for the (hyper)operads governing modular, cyclic and ordinary operads, and wheeled properads, respectively. Algebras for these models are homotopy versions of the corresponding structures.

The independent domination number $i(G)$ of a graph $G$ is the minimum cardinality of a maximal independent set of $G$, also called an $i(G)$-set. The $i$-graph of $G$, denoted $\mathscr{I}(G)$, is the graph whose vertices correspond to the…

Let $G$ be a simple, undirected graph on the vertex set $V=\{1,2,\ldots ,n\}$ and let $A$ be the adjacency matrix of $G.$ A non-empty subset $ \{i_{1},i_{2},\ldots ,i_{k}\}$ of $V$ is called a driver set for $G$ if the system…

We study separating systems of the edges of a graph where each member of the separating system is a path. We conjecture that every $n$-vertex graph admits a separating path system of size $O(n)$ and prove this in certain interesting special…

A road interchange where $n$ roads meet and in which the drivers are not allowed to change lanes can be modelled as an embedding of a 2-coloured (hence bipartite) multigraph $G$ with equal-sized colour classes into an orientable surface…

For an acyclic directed graph with multiple sources and multiple sinks, we prove that one can choose the Merger's paths between the sources and the sinks such that the number of mergings between these paths is upper bounded by a constant…

This is my dissertation about digraphs ordered by pp-constructability. We study in particular smooth digraphs, i.e., digraphs without sources or sinks, tournaments and semicomplete digraphs, orientations of paths and cycles, digraphs with…

If $H$ is (or is isomorphic to) a subgraph of $G$, $H$ is said to {\it divide} $G$ if there is an edge-decomposition of $G$ by copies of $E(H)$, the edge set of $H$. A more restrictive version of this is when there is a subgroup ${\cal H}$…

The homological scaffold leverages persistent homology to construct a topologically sound summary of a weighted network. However, its crucial dependency on the choice of representative cycles hinders the ability to trace back global…

A modified version of the Dijkstra algorithm using an inventive contraction hierarchy is proposed. The algorithm considers a directed acyclic graph with a conical or semi-circular structure for which a pair of edges is chosen iteratively…

We study the least doubling constant $C_G$ among all possible doubling measures defined on a path graph $G$. We consider both finite and infinite cases and show that, if $G=\mathbb Z$, $C_{\mathbb Z}=3$, while for $G=L_n$, the path graph…