The minimum number of edges in graphs with prescribed paths

Minimum number of edges in graphs that are both P2- and Pi-connected

A graph G is k-critical if G is not (k − 1)-colorable, but every proper subgraph of G is (k − 1)-colorable. A graph G is k-choosable if G has an L-coloring from every list assignment L with |L(v)|=k for all v, and a graph G is k-list-critical if G is not (k−1)-choosable, but every proper subgraph of G is (k−1)-choosable. The problem of determining the minimum number of edges in a k-critical graph with n vertices has been widely studied, starting with work of Gallai and culminating with the seminal results of Kostochka and Yancey, who essentially solved the problem. In this paper, we improve the best known lower bound on the number of edges in a k-list-critical graph. In fact, our result on k-list-critical graphs is derived from a lower bound on the number of edges in a graph with Alon–Tarsi number at least k. Our proof uses the discharging method, which makes it simpler and more modular than previous work in this area.

A number of modeling and simulation algorithms using internal coordinates rely on hierarchical representations of molecular systems. Given the potentially complex topologies of molecular systems, though, automatically generating such hierarchical decompositions may be difficult. In this article, we present a fast general algorithm for the complete construction of a hierarchical representation of a molecular system. This two-step algorithm treats the input molecular system as a graph in which vertices represent atoms or pseudo-atoms, and edges represent covalent bonds. The first step contracts all cycles in the input graph. The second step builds an assembly tree from the reduced graph. We analyze the complexity of this algorithm and show that the first step is linear in the number of edges in the input graph, whereas the second one is linear in the number of edges in the graph without cycles, but dependent on the branching factor of the molecular graph. We demonstrate the performance of our algorithm on a set of specifically tailored difficult cases as well as on a large subset of molecular graphs extracted from the protein data bank. In particular, we experimentally show that both steps behave linearly in the number of edges in the input graph (the branching factor is fixed for the second step). Finally, we demonstrate an application of our hierarchy construction algorithm to adaptive torsion-angle molecular mechanics.

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings. As a generalization, Liu and Liu recently introduced the concept of fractional matching preclusion number. The fractional matching preclusion number (FMP number) of [Formula: see text], denoted by [Formula: see text], is the minimum number of edges whose deletion leaves the resulting graph without a fractional perfect matching. The fractional strong matching preclusion number (FSMP number) of [Formula: see text], denoted by [Formula: see text], is the minimum number of vertices and edges whose deletion leaves the resulting graph without a fractional perfect matching. In this paper, we study the fractional matching preclusion number and the fractional strong matching preclusion number for the radix triangular mesh [Formula: see text], and all the optimal fractional matching preclusion sets and fractional strong matching preclusion sets of these graphs are categorized.

NP-hardness of multiple bondage in graphs

Fractional matching preclusion for arrangement graphs

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings. As a generalization, Liu and Liu introduced the concept of fractional matching preclusion number in 2017. The Fractional Matching Preclusion Number (FMP number) of G is the minimum number of edges whose deletion leaves the resulting graph without a fractional perfect matching. The Fractional Strong Matching Preclusion Number (FSMP number) of G is the minimum number of vertices and/or edges whose deletion leaves the resulting graph without a fractional perfect matching. In this paper, we obtain the FMP number and the FSMP number for (n, k)-star graphs. In addition, all the optimal fractional strong matching preclusion sets of these graphs are categorized.

The matching preclusion number of graph G is the minimum number of edges whose deletion leaves the resulting graph without a perfect matching or almost-perfect matching. The strong matching preclusion number of a graph G is the minimum number of vertices and edges whose deletion leaves the resulting graph without a perfect matching or an almost-perfect matching. The conditional matching preclusion number of G is the minimum number of edges whose deletion leaves the resulting graph with no isolated vertices and without a perfect matching or almost-perfect matching. In this paper, we study the matching preclusion number of radix triangular mesh with an odd number of vertices, and strong matching preclusion number and conditional matching preclusion number of radix triangular mesh. Also, we obtained the radix triangular mesh with an even number of vertices is super strongly matched and conditionally super matched.

We study the task of estimating the number of edges in a graph, where the access to the graph is provided via an independent set oracle. Independent set queries draw motivation from group testing and have applications to the complexity of decision versus counting problems. We give two algorithms to estimate the number of edges in an n -vertex graph, using (i) polylog( n ) bipartite independent set queries or (ii) n 2/3 polylog( n ) independent set queries.

Extremal graphs for blow-ups of stars and paths

The formula for Turán number of spanning linear forests

Covering Non-uniform Hypergraphs

Constructing extremal triangle-free graphs using integer programming

Strong matching preclusion number of graphs

Given graphs $G$ and $H$, $G$ is $H$-saturated if $H$ is not a subgraph of $G$, but for all $e \notin E(G)$, $H$ appears as a subgraph of $G + e$. While for every $n \ge |V(H)|$, there exists an $n$-vertex graph that is $H$-saturated, the same does not hold for induced subgraphs. That is, there exist graphs $H$ and values of $n \ge |V(H)|$, for which every $n$-vertex graph $G$ either contains $H$ as an induced subgraph, or there exists $e \notin E(G)$ such that $G + e$ does not contain $H$ as an induced subgraph. To circumvent this Martin and Smith make use of a generalized notion of "graph" when introducing the concept of induced saturation and the induced saturation number of graphs. This allows for edges that can be included or excluded when searching for an induced copy of $H$, and the induced saturation number is the minimum number of such edges that are required.In this paper, we show that the induced saturation number of many common graphs is zero. This yields graphs that are $H$-induced-saturated. That is, graphs such that no induced copy of $H$ exists, but adding or deleting any edge creates an induced copy of $H$. We introduce a new parameter for such graphs, indsat*($n;H$), which is the minimum number of edges in an $H$-induced-saturated graph. We provide bounds on indsat*($n;H$) for many graphs. In particular, we determine indsat*($n;H$) completely when $H$ is the paw graph $K_{1,3}+e$, and we determine indsat*(n;$K_{1,3}$) within an additive constant of four.