A set of vertices in a graph G is a global dominating set of G if it dominates both G and its complement G. The minimum cardinality of a global dominating set of G is the global domination number of G. We explore the effects of graph modifications (edge removal, vertex removal, and edge addition) on the global domination number. In particular, for each graph modification, we study the global domination stable trees, that is, the trees whose global domination number remains the same upon the modification. We characterize these stable trees having small global domination numbers.

The complementary prism GḠ of a graph G is formed from the disjoint union of G and its complement Ḡ by adding the edges of a perfect matching between the corresponding vertices of G and Ḡ. A set S ⊆ V(G) is a double dominating set of G if for every v ∈ V(G)\S, v is adjacent to at least two vertices of S, and for every w ∈ S, w is adjacent to at least one vertex of S. The double domination number of G is the minimum cardinality of a double dominating set of G. We begin by determining the double domination number of complementary prisms of paths and cycles. Then we characterize the graphs G whose complementary prisms have small double domination numbers. Finally, we establish lower and upper bounds on the double domination number of GḠ and show that all values between these bounds are attainable.

A set of vertices in a graph G is a global dominating set of G if it dominates both G and its complement G. The minimum cardinality of a global dominating set of G is the global domination number of G. We explore the effects of graph modifications (edge removal, vertex removal, and edge addition) on the global domination number. In particular, for each graph modification, we study the global domination stable trees, that is, the trees whose global domination number remains the same upon the modification. We characterize these stable trees having small global domination numbers.

A total dominating set in a graph G is a set S of vertices of G such that every vertex in G is adjacent to a vertex of S. We study graphs whose vertex set can be partitioned into two total dominating sets. In particular, we develop several sufficient conditions for a graph to have a vertex partition into two total dominating sets. We also show that with the exception of the cycle on five vertices, every selfcomplementary graph with minimum degree at least two has such a partition.

A graph G is diameter 2-critical if its diameter is two and the deletion of any edge increases the diameter. Murty and Simon conjectured that the number of edges in a diameter 2-critical graph of order n is at most n2/4 and that the extremal graphs are complete bipartite graphs with equal size partite sets. We use an important association with total domination to prove the conjecture for the graphs whose complements have vertex connectivity k for k∈1,2,3.

A graph G is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. In this paper we characterize the diameter-2-critical graphs with no antihole of length four, that is, the diameter-2-critical graphs whose complements have no induced 4-cycle. Murty and Simon conjectured that the number of edges in a diameter-2-critical graph of order n is at most n 2/4 and that the extremal graphs are complete bipartite graphs with equal size partite sets. As a consequence of our characterization, we prove the Murty-Simon Conjecture for graphs with no antihole of length four.

The complementary prism GḠ of a graph G is formed from the disjoint union of G and its complement G by adding the edges of a perfect matching between the corresponding vertices of G and Ḡ. A set S ⊆ V(G) is a restrained dominating set of G if for every v € V(G) \S, v is adjacent to a vertex in S and a vertex in V(G) \S. The restrained domination number of G is the minimum cardinality of a restrained dominating set of G. We study restrained domination of complementary prisms. In particular, we establish lower and upper bounds on the restrained domination number of GḠ, show that the restrained domination number can be attained for all values between these bounds, and characterize the graphs which attain the lower bound.

For a graph G = (V, E), a non-empty set S ⊆ V is a defensive alliance if for every vertex v in S, v has at most one more neighbor in V - S than it has in S, and S is an offensive alliance if for every v ∈ V - S that has a neighbor in S, v has more neighbors in S than in V - S. A powerful alliance is both defensive and offensive. We initiate the study of powerful alliances in graphs.

A total dominating set in a graph G is a set S of vertices of G such that every vertex in G is adjacent to a vertex of S. We study cubic graphs whose vertex set can be partitioned into two total dominating sets. There are infinitely many examples of connected cubic graphs that do not have such a vertex partition. In this paper, we show that the class of claw-free cubic graphs has such a partition. For an integer k at least 3, a graph is k-chordal if it does not have an induced cycle of length more than k. Chordal graphs coincide with 3-chordal graphs. We observe that for k≥6, not every graph in the class of k-chordal, connected, cubic graphs has two vertex disjoint total dominating sets. We prove that the vertex set of every 5-chordal, connected, cubic graph can be partitioned into two total dominating sets. As a consequence of this result, we observe that this property also holds for a connected, cubic graph that is chordal or 4-chordal. We also prove that cubic graphs containing a diamond as a subgraph can be partitioned into two total dominating sets.

The purpose of this essay is to investigate the importance of gender in the security guard occupation and also to investigate what gender-marked notions about the security industry that exist among the respondents. Furthermore, the purpose is to investigate the respondents' experience of the impact that gender has on the working group. Previous research on police officers shows that within the police force there are perceptions of how one should be and act that are related to notions of the biological sex. Furthermore, research shows that the ideal of masculinity is central to the profession, as there are preconceived notions that a police officer needs to have certain masculine qualities. Through six qualitative semi-structured interviews, we examined the respondents' experience of the importance of gender in their work as security guards as well as for the working group. We also examined the respondents' perceptions of the industry. The results show that all respondents have a positive image of the profession, where they are treated with respect regardless of gender. Some respondents describe that each gender has certain characteristics that distinguish them, but that it does not affect how well they can perform their work. Furthermore, it appears that all respondents' see the profession of security guards as a masculine industry. On the other hand, it is not masculine qualities that are described that are necessary in the profession. Rather softer qualities such as empathy are described as necessary. / Syftet med denna uppsats är att undersöka vilken betydelse som väktare och ordningsvakter upplever att kön har i deras arbete och hur det påverkar arbetsgruppen, samt undersöka vilka genusmärkta föreställningar som finns bland respondenterna om branschen i sin helhet. Tidigare forskning om poliser visar att det inom poliskåren finns uppfattningar om hur man bör vara och agera som är kopplade till föreställningar om det biologiska könet. Vidare visar forskningen att maskulinitetsideal är centralt inom yrket, då det finns förutfattade meningar om att en polis behöver ha vissa maskulina egenskaper. Genom sex kvalitativa semistrukturerade intervjuer har vi undersökt respondenternas upplevelse av betydelsen av kön i arbetet som ordningsvakt och väktare, liksom köns inverkan på arbetsgruppen samt undersöka respondenternas föreställningar om branschen. Resultatet visar att samtliga respondenter har en positiv bild av yrket, där man behandlas med respekt oavsett kön. Vissa respondenter beskriver att respektive kön har vissa egenskaper som utmärker dem, men att det inte är något som påverkar hur väl de kan utföra sitt arbete. Vidare framkommer det också att samtliga respondenter sett branschen som maskulint förknippad, men när respondenterna beskriver de egenskaperna som anses som nödvändiga och fördelaktiga i yrket är det snarare mjukare egenskaper som beskrivs exempelvis inkännande och empati som nämns.