相关论文: Families of $2$-weights of some particular graphs

Let ${\cal G}=(G,w)$ be a weighted simple finite connected graph, that is, let $G$ be a simple finite connected graph endowed with a function $w$ from the set of the edges of $G$ to the set of real numbers. For any subgraph $G'$ of $G$, we…

Let ${\cal G}=(G,w)$ be a weighted graph , that is, a graph $G$ endowed with a function $w$ from the edge set of $G$ to the set of real numbers; for any subset $S$ of the vertex set of $G$, we define $D_S({\cal G})$ to be the minimum of the…

Let {\cal G}=(G,w) be a positive-weighted simple finite graph, that is, let G be a simple finite graph endowed with a function w from the set of the edges of G to the set of the positive real numbers. For any subgraph G' of G, we define…

Consider a weighted graph G with n vertices, numbered by the set {1,...,n}. For any path p in G, we call w_G(p) the sum of the weights of the edges of the path and we define the multiset {\cal D}_{i,j} (G) = {w_G(p) | p simple path between…

Let ${\cal T}=(T,w)$ be a weighted finite tree with leaves $1,..., n$.For any $I :=\{i_1,..., i_k \} \subset \{1,...,n\}$, let $D_I ({\cal T})$ be the weight of the minimal subtree of $T$ connecting $i_1,..., i_k$; the $D_{I} ({\cal T})$…

Let $G$ be a graph, and let $w$ be a positive real-valued weight function on $V(G)$. For every subset $S$ of $V(G)$, let $w(S)=\sum_{v \in S} w(v).$ A non-empty subset $S \subset V(G)$ is a weighted safe set of $(G,w)$ if, for every…

A {\it semi-proper orientation} of a given graph $G$ is a function $(D,w)$ that assigns an orientation $D(e)$ and a positive integer weight $ w(e)$ to each edge $e$ such that for every two adjacent vertices $v$ and $u$, $S_{(D,w)}(v) \neq…

Let ${\cal T}=(T,w)$ be a weighted finite tree with leaves $1,..., n$. For any $I :=\{i_1,..., i_k \} \subset \{1,...,n\}$, let $D_I ({\cal T})$ be the weight of the minimal subtree of $T$ connecting $i_1,..., i_k$; the $D_{I} ({\cal T})$…

A {\it star-factor} of a graph $G$ is a spanning subgraph of $G$ such that each component of which is a star. An {\it edge-weighting} of $G$ is a function $w: E(G)\longrightarrow \mathbb{N}^+$, where $\mathbb{N}^+$ is the set of positive…

We study "positive" graphs that have a nonnegative homomorphism number into every edge-weighted graph (where the edgeweights may be negative). We conjecture that all positive graphs can be obtained by taking two copies of an arbitrary…

We study a family of positive weighted well-covered graphs, which we call levelable graphs, that are related to a construction of level artinian rings in commutative algebra. A graph $G$ is levelable if there exists a weight function with…

Let ${\cal T}=(T,w)$ be a positive-weighted tree with at least $n$ vertices. For any $i,j \in \{1,...,n\}$, let $D_{i,j} ({\cal T})$ be the weight of the unique path in $T$ connecting $i$ and $j$. The $D_{i,j} ({\cal T})$ are called…

Let ${\cal T}=(T,w)$ be a weighted finite tree with leaves $1,..., n$.For any $I :=\{i_1,..., i_k \} \subset \{1,...,n\}$,let $D_I ({\cal T})$ be the weight of the minimal subtree of $T$ connecting $i_1,..., i_k$; the $D_{I} ({\cal T})$ are…

Let $G$ be a graph, and let $w: V(G) \to \mathbb{R}$ be a weight function on the vertices of $G$. For every subset $X$ of $V(G)$, let $w(X)=\sum_{v \in X} w(v).$ A non-empty subset $S \subset V(G)$ is a weighted safe set of $(G,w)$ if, for…

Given a connected graph $G=(V,E)$ and a crossing family $\mathcal{C}$ over ground set $V$ such that $|\delta_G(U)|\geq 2$ for every $U\in \mathcal{C}$, we prove there exists a strong orientation of $G$ for $\mathcal{C}$, i.e., an…

A weighted graph is a graph in which every edge is assigned a non-negative real number. In a weighted graph, the weight of a path is the sum of the weights of its edges, and the weighed degree of a vertex is the sum of the weights of the…

Let ${\mathcal G}$ be an infinite family of connected graphs and let $k$ be a positive integer. We say that $k$ is ${\it forcing}$ for ${\mathcal G}$ if for all $G \in {\mathcal G}$ but finitely many, the following holds. Any…

A graph $G$ is $(k,k')$-choosable if the following holds: For any list assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ real numbers, and assigns to each edge $e$ a set $L(e)$ of $k'$ real numbers, there is a total…

For a graph $G$, two dominating sets $D$ and $D'$ in $G$, and a non-negative integer $k$, the set $D$ is said to $k$-transform to $D'$ if there is a sequence $D_0,\ldots,D_\ell$ of dominating sets in $G$ such that $D=D_0$, $D'=D_\ell$,…

A graph $G$ is a $D\!D_2$-graph if it has a pair $(D,D_2)$ of disjoint sets of vertices of $G$ such that $D$ is a dominating set and $D_2$ is a 2-dominating set of $G$. We provide several characterizations and hardness results concerning…