A set D of vertices in a graph G is a dominating set if every vertex in V −D is adjacent to some vertex in D. The domination number γ(G) is the minimum cardinality of a dominating set of G. A dominating set D of a graph G is total dominating set if the induced subgraph 〈D〉 has no isolated vertices. In this paper, we introduce the total co-independent domination in graphs, exact value for some s...

Let $kgeq 1$ be an integer, and let $G$ be a graph. A {it$k$-rainbow dominating function} (or a {it $k$-RDF}) of $G$ is afunction $f$ from the vertex set $V(G)$ to the family of all subsetsof ${1,2,ldots ,k}$ such that for every $vin V(G)$ with$f(v)=emptyset $, the condition $bigcup_{uinN_{G}(v)}f(u)={1,2,ldots,k}$ is fulfilled, where $N_{G}(v)$ isthe open neighborhood of $v$. The {it weight} o...

A graph is well-indumatched if all its maximal induced matchings are of the same size. We first prove that recognizing the class WIM of well-indumatched graphs is a co-NPcomplete problem even for (2P5,K1,5)-free graphs. We then show that the well-known decision problems such as Independent Dominating Set, Independent Set, and Dominating Set are NP-complete for well-indumatched graphs. We also s...

In this paper, we define a new domination-like invariant of graphs. Let $${\mathbb {R}}^{+}$$ be the set non-negative numbers. $$c\in {\mathbb {R}}^{+}-\{0\}$$ number, and let G graph. A function $$f:V(G)\rightarrow is c-self-dominating if for every $$u\in V(G)$$ , $$f(u)\ge c$$ or $$\max \{f(v):v\in N_{G}(u)\}\ge 1$$ . The c-self-domination number $$\gamma ^{c}(G)$$ defined as ^{c}(G):=\min \{...

We study, for a countably categorical theory T, the complexity of computing and the complexity of dominating the function specifying the number of n-types consistent with T.

Acknowledgements I would like to express my deepest appreciation to my supervisors Professor Hans-Jürgen Schmeisser and Professor Winfried Sickel for their support and many hints and comments. I thank also Professor Hans Triebel for many valuable discussions on the topic of this work.

<abstract><p>Let $ G be a graph with vertex set V(G) $. A function f:V(G)\rightarrow \{0, 1, 2\} is Roman dominating on if every v\in for which f(v) = 0 adjacent to at least one u\in such that f(u) 2 The domination number of the minimum weight \omega(f) \sum_{x\in V(G)}f(x) among all functions f In this article we study direct product graphs and rooted graphs. Specifically, give sev...

Let G be a graph with no isolated vertex and let N(v) the open neighbourhood of v∈V(G). f:V(G)→{0,1,2} function Vi={v∈V(G):f(v)=i} for every i∈{0,1,2}. We say that f is strongly total Roman dominating on if subgraph induced by V1∪V2 has N(v)∩V2≠∅ v∈V(G)\V2. The domination number G, denoted γtRs(G), defined as minimum weight ω(f)=∑x∈V(G)f(x) among all functions G. This paper devoted to study it ...