﻿ Lutz Volkmann

# Lutz Volkmann

RWTH Aachen University

## [ 1 ] - Lower bounds on the signed (total) \$k\$-domination number

Let \$G\$ be a graph with vertex set \$V(G)\$. For any integer \$kge 1\$, a signed (total) \$k\$-dominating functionis a function \$f: V(G) rightarrow { -1, 1}\$ satisfying \$sum_{xin N[v]}f(x)ge k\$ (\$sum_{xin N(v)}f(x)ge k\$)for every \$vin V(G)\$, where \$N(v)\$ is the neighborhood of \$v\$ and \$N[v]=N(v)cup{v}\$. The minimum of the values\$sum_{vin V(G)}f(v)\$, taken over all signed (total) \$k\$-dominating functi...

## [ 2 ] - Double Roman domination and domatic numbers of graphs

A double Roman dominating function on a graph \$G\$ with vertex set \$V(G)\$ is defined in cite{bhh} as a function\$f:V(G)rightarrow{0,1,2,3}\$ having the property that if \$f(v)=0\$, then the vertex \$v\$ must have at least twoneighbors assigned 2 under \$f\$ or one neighbor \$w\$ with \$f(w)=3\$, and if \$f(v)=1\$, then the vertex \$v\$ must haveat least one neighbor \$u\$ with \$f(u)ge 2\$. The weight of a double R...

## [ 3 ] - Sufficient conditions for maximally edge-connected and super-edge-connected

Let \$G\$ be a connected graph with minimum degree \$delta\$ and edge-connectivity \$lambda\$. A graph ismaximally edge-connected if \$lambda=delta\$, and it is super-edge-connected if every minimum edge-cut istrivial; that is, if every minimum edge-cut consists of edges incident with a vertex of minimum degree.In this paper, we show that a connected graph or a connected triangle-free graph is maximall...

## [ 4 ] - Signed total Roman k-domination in directed graphs

Let \$D\$ be a finite and simple digraph with vertex set \$V(D)\$‎.‎A signed total Roman \$k\$-dominating function (STR\$k\$DF) on‎‎\$D\$ is a function \$f:V(D)rightarrow{-1‎, ‎1‎, ‎2}\$ satisfying the conditions‎‎that (i) \$sum_{xin N^{-}(v)}f(x)ge k\$ for each‎‎\$vin V(D)\$‎, ‎where \$N^{-}(v)\$ consists of all vertices of \$D\$ from‎‎which arcs go into \$v\$‎, ‎and (ii) every vertex \$u\$ for which‎‎\$f(u)=-1\$ has a...

## [ 5 ] - Sufficient conditions on the zeroth-order general Randic index for maximally edge-connected digraphs

Let D be a digraph with vertex set V(D) .For vertex v V(D), the degree of v, denoted by d(v), is defined as the minimum value if its out-degree  and its in-degree . Now let D be a digraph with minimum degree  and edge-connectivity If  is real number, then the zeroth-order general Randic index is defined by   .  A digraph is maximally edge-connected if . In this paper we present sufficient condi...

## [ 6 ] - The Italian domatic number of a digraph

An {em Italian dominating function} on a digraph \$D\$ with vertex set \$V(D)\$ is defined as a function\$fcolon V(D)to {0, 1, 2}\$ such that every vertex \$vin V(D)\$ with \$f(v)=0\$ has at least two in-neighborsassigned 1 under \$f\$ or one in-neighbor \$w\$ with \$f(w)=2\$. A set \${f_1,f_2,ldots,f_d}\$ of distinctItalian dominating functions on \$D\$ with the property that \$sum_{i=1}^d f_i(v)le 2\$ for each \$vi...

## [ 7 ] - A note on the Roman domatic number of a digraph

Roman dominating function} on a digraph \$D\$ with vertex set \$V(D)\$ is a labeling\$fcolon V(D)to {0, 1, 2}\$such that every vertex with label \$0\$ has an in-neighbor with label \$2\$. A set \${f_1,f_2,ldots,f_d}\$ ofRoman dominating functions on \$D\$ with the property that \$sum_{i=1}^d f_i(v)le 2\$ for each \$vin V(D)\$,is called a {em Roman dominating family} (of functions) on \$D\$....

## [ 8 ] - Total double Roman domination in graphs

Let \$G\$ be a simple graph with vertex set \$V\$. A double Roman dominating function (DRDF) on \$G\$ is a function \$f:Vrightarrow{0,1,2,3}\$ satisfying that if \$f(v)=0\$, then the vertex \$v\$ must be adjacent to at least two vertices assigned \$2\$ or one vertex assigned \$3\$ under \$f\$, whereas if \$f(v)=1\$, then the vertex \$v\$ must be adjacent to at least one vertex assigned \$2\$ or \$3\$. The weight of a DR...

## [ 9 ] - Weak signed Roman domination in graphs

A {em weak signed Roman dominating function} (WSRDF) of a graph \$G\$ with vertex set \$V(G)\$ is defined as afunction \$f:V(G)rightarrow{-1,1,2}\$ having the property that \$sum_{xin N[v]}f(x)ge 1\$ for each \$vin V(G)\$, where \$N[v]\$ is theclosed neighborhood of \$v\$. The weight of a WSRDF is the sum of its function values over all vertices.The weak signed Roman domination number of \$G...

## [ 10 ] - Nonnegative signed total Roman domination in graphs

‎Let \$G\$ be a finite and simple graph with vertex set \$V(G)\$‎. ‎A nonnegative signed total Roman dominating function (NNSTRDF) on a‎ ‎graph \$G\$ is a function \$f:V(G)rightarrow{-1‎, ‎1‎, ‎2}\$ satisfying the conditions‎‎that (i) \$sum_{xin N(v)}f(x)ge 0\$ for each‎ ‎\$vin V(G)\$‎, ‎where \$N(v)\$ is the open neighborhood of \$v\$‎, ‎and (ii) every vertex \$u\$ for which‎ ‎\$f(u...

## [ 11 ] - Weak signed Roman k-domination in graphs

Let \$kge 1\$ be an integer, and let \$G\$ be a finite and simple graph with vertex set \$V(G)\$.A weak signed Roman \$k\$-dominating function (WSRkDF) on a graph \$G\$ is a function\$f:V(G)rightarrow{-1,1,2}\$ satisfying the conditions that \$sum_{xin N[v]}f(x)ge k\$ for eachvertex \$vin V(G)\$, where \$N[v]\$ is the closed neighborhood of \$v\$. The weight of a WSRkDF \$f\$ is\$w(f)=sum_{vin V(G)}f(v)\$. The weak si...

## [ 12 ] - Bounds on the outer-independent double Italian domination number

An outer-independent double Italian dominating function (OIDIDF)on a graph \$G\$ with vertex set \$V(G)\$ is a function\$f:V(G)longrightarrow {0,1,2,3}\$ such that if \$f(v)in{0,1}\$ for a vertex \$vin V(G)\$ then \$sum_{uin N[v]}f(u)geq3\$,and the set \$ {uin V(G)|f(u)=0}\$ is independent. The weight ofan OIDIDF \$f\$ is the value \$w(f)=sum_{vin V(G)}f(v)\$. Theminimum weight of an OIDIDF on a graph \$G\$ is cal...

## [ 13 ] - Signed total Italian k-domination in graphs

Let k ≥ 1 be an integer, and let G be a finite and simple graph with vertex set V (G). A signed total Italian k-dominating function (STIkDF) on a graph G is a functionf : V (G) → {−1, 1, 2} satisfying the conditions that \$sum_{xin N(v)}f(x)ge k\$ for each vertex v ∈ V (G), where N(v) is the neighborhood of \$v\$, and each vertex u with f(u)=-1 is adjacent to a vertex v with f(v)=2 or to two vertic...

نویسندگان همکار