Abstract: Several papers (e.g. [7, 23, 42]) have recently sought to give general frameworks for online structures and algorithms ([4]), and seeking to connect, if only by analogy, online and computable structure theory. These initiatives build on earlier work on online colouring and other combinatorial algorithms by Bean [10], Kierstead, Trotter et al. [48, 54, 57] and others, as we discuss below. In this paper we will look at such frameworks and illustrate them with examples from the first author’s MSc Thesis ([58]). In this thesis, Askes looks at online, adversarial online and strongly online algorithms and structures. Additionally, we prove some new theorems about online algorithms on graphs of bounded pathwidth as well as some new results on punctually 1-decidable structures.

Abstract: In this thesis we explore a variety of online and adversarial algorithms. We primarily explore the following online and adversarial algorithms; the perfect code game, adversarial online colouring, the chain decomposition game, and strongly online graphs.

The perfect code game is a new adversarial game played on graphs in which players take turns constructing perfect codes. We provide definitions for perfect codes and the perfect code game, along with some motivation from coding theory. We will prove upper bounds for both cycle and path graphs. We also prove an upper bound for graphs of bounded pathwidth. Finally, we explore the perfect code game in graphs of bounded degree.

We will create a new game called the adversarial online colouring game, this game takes elements from both online and adversarial algorithms. We will begin with some discussion and a definition of adversarial online colouring. We will then prove several results related to graph degree. We conclude with a proof that the adversarial online colouring game on trees is determined only by the number of colours and the number of vertices (assuming that both players have a chance at winning).

The chain decomposition game is another adversarial game, but this time played on partial orders. We introduce the chain decomposition game and demonstrate two results relating to upper and lower bounds for the game. We also prove a result on the online adversarial version of the game.

We introduce strongly online graphs and graph colouring as a new algorithmic parameterization to online graph colouring. A strongly online graph is an online graph where at each stage 𝑠 we can see a ball of increasing radius about each vertex. We will prove several bounds (upper and lower) on the online chromatic number of strongly online graphs. For example, we show that every strongly online graph can be coloured in twice its chromatic number. We prove that every strongly online graph with even pathwidth 𝑘 can be online coloured with 2𝑘 + 1 colours. Then, after introducing a natural notion of strongly online pathwidth, we prove that there is a strongly online graph with no finite strongly online path decomposition.

Abstract: A graph algorithm with a hostile partner is a game played between two players, Alice and Bob. In each game Alice and Bob take it in turns to construct some object. Alice wins if the object has a specific property, and Bob wins if it doesn't. In this report we will explore a variety of games along with variations that allow Alice and Bob to play more than once per turn. We will also examine various strategies for both Alice and Bob. And, for each strategy we will see under what conditions Alice or Bob will always win. The four main games we will consider are; the dominating game, the independent dominating game, the colouring game, and the marking game.