Colouring Diamond-free Graphs

The Colouring problem is that of deciding, given a graph G and an integer k, whether G admits a (proper) k-colouring. For all graphs H up to five vertices, we classify the computational complexity of Colouring for (diamond,H)-free graphs. Our proof is based on combining known results together with proving that the clique-width is bounded for (diamond,P_1+2P_2)-free graphs. Our technique for handling this case is to reduce the graph under consideration to a k-partite graph that has a very specific decomposition. As a by-product of this general technique we are also able to prove boundedness of clique-width for four other new classes of (H_1,H_2)-free graphs. As such, our work also continues a recent systematic study into the (un)boundedness of clique-width of (H_1,H_2)-free graphs, and our five new classes of bounded clique-width reduce the number of open cases from 13 to 8

Structural solutions to maximum independent set and related problems

In this thesis, we study some fundamental problems in algorithmic graph theory. Most natural problems in this area are hard from a computational point of view. However, many applications demand that we do solve such problems, even if they are intractable. There are a number of methods in which we can try to do this: 1) We may use an approximation algorithm if we do not necessarily require the best possible solution to a problem. 2) Heuristics can be applied and work well enough to be useful for many applications. 3) We can construct randomised algorithms for which the probability of failure is very small. 4) We may parameterize the problem in some way which limits its complexity. In other cases, we may also have some information about the structure of the instances of the problem we are trying to solve. If we are lucky, we may and that we can exploit this extra structure to find efficient ways to solve our problem. The question which arises is - How far must we restrict the structure of our graph to be able to solve our problem efficiently? In this thesis we study a number of problems, such as Maximum Indepen- dent Set, Maximum Induced Matching, Stable-II, Efficient Edge Domina- tion, Vertex Colouring and Dynamic Edge-Choosability. We try to solve problems on various hereditary classes of graphs and analyse the complexity of the resulting problem, both from a classical and parameterized point of view

Playing with parameters: structural parameterization in graphs

When considering a graph problem from a parameterized point of view, the parameter chosen is often the size of an optimal solution of this problem (the "standard" parameter). A natural subject for investigation is what happens when we parameterize such a problem by various other parameters, some of which may be the values of optimal solutions to different problems. Such research is known as parameterized ecology. In this paper, we investigate seven natural vertex problems, along with their respective parameters: the size of a maximum independent set, the size of a minimum vertex cover, the size of a maximum clique, the chromatic number, the size of a minimum dominating set, the size of a minimum independent dominating set and the size of a minimum feedback vertex set. We study the parameterized complexity of each of these problems with respect to the standard parameter of the others.Comment: 17 page

Finding a Small Number of Colourful Components

A partition (V_1,...,V_k) of the vertex set of a graph G with a (not necessarily proper) colouring c is colourful if no two vertices in any V_i have the same colour and every set V_i induces a connected graph. The Colourful Partition problem, introduced by Adamaszek and Popa, is to decide whether a coloured graph (G,c) has a colourful partition of size at most k. This problem is related to the Colourful Components problem, introduced by He, Liu and Zhao, which is to decide whether a graph can be modified into a graph whose connected components form a colourful partition by deleting at most p edges. Despite the similarities in their definitions, we show that Colourful Partition and Colourful Components may have different complexities for restricted instances. We tighten known NP-hardness results for both problems by closing a number of complexity gaps. In addition, we prove new hardness and tractability results for Colourful Partition. In particular, we prove that deciding whether a coloured graph (G,c) has a colourful partition of size 2 is NP-complete for coloured planar bipartite graphs of maximum degree 3 and path-width 3, but polynomial-time solvable for coloured graphs of treewidth 2. Rather than performing an ad hoc study, we use our classical complexity results to guide us in undertaking a thorough parameterized study of Colourful Partition. We show that this leads to suitable parameters for obtaining FPT results and moreover prove that Colourful Components and Colourful Partition may have different parameterized complexities, depending on the chosen parameter

Editing to Eulerian Graphs

We investigate the problem of modifying a graph into a connected graph in which the degree of each vertex satisfies a prescribed parity constraint. Let ea, ed and vd denote the operations edge addition, edge deletion and vertex deletion respectively. For any S subseteq {ea,ed,vd}, we define Connected Degree Parity Editing (S) (CDPE(S)) to be the problem that takes as input a graph G, an integer k and a function delta: V(G) -> {0,1}, and asks whether G can be modified into a connected graph H with d_H(v) = delta(v)(mod 2) for each v in V(H), using at most k operations from S. We prove that (*) if S={ea} or S={ea,ed}, then CDPE(S) can be solved in polynomial time; (*) if {vd} subseteq S subseteq {ea,ed,vd}, then CDPE(S) is NP-complete and W-hard when parameterized by k, even if delta = 0. Together with known results by Cai and Yang and by Cygan, Marx, Pilipczuk, Pilipczuk and Schlotter, our results completely classify the classical and parameterized complexity of the CDPE(S) problem for all S subseteq {ea,ed,vd}. We obtain the same classification for a natural variant of the cdpe(S) problem on directed graphs, where the target is a weakly connected digraph in which the difference between the in- and out-degree of every vertex equals a prescribed value. As an important implication of our results, we obtain polynomial-time algorithms for Eulerian Editing problem and its directed variant. To the best of our knowledge, the only other natural non-trivial graph class H for which the H-Editing problem is known to be polynomial-time solvable is the class of split graphs.publishedVersio

Clique‐width: Harnessing the power of atoms

Many NP-complete graph problems are polynomial-time solvable on graph classes of bounded clique-width. Several of these problems are polynomial-time solvable on a hereditary graph class if they are so on the atoms (graphs with no clique cut-set) of . Hence, we initiate a systematic study into boundedness of clique-width of atoms of hereditary graph classes. A graph is -free if is not an induced subgraph of , and it is -free if it is both -free and -free. A class of -free graphs has bounded clique-width if and only if its atoms have this property. This is no longer true for -free graphs, as evidenced by one known example. We prove the existence of another such pair and classify the boundedness of clique-width on -free atoms for all but 18 cases

Editing to Eulerian Graphs

We investigate the problem of modifying a graph into a connected graph in which the degree of each vertex satisfies a prescribed parity constraint. Let $ea$, $ed$ and $vd$ denote the operations edge addition, edge deletion and vertex deletion respectively. For any $S\subseteq \{ea,ed,vd\}$, we define Connected Degree Parity Editing$(S)$ (CDPE($S$)) to be the problem that takes as input a graph $G$, an integer $k$ and a function $\delta\colon V(G)\rightarrow\{0,1\}$, and asks whether $G$ can be modified into a connected graph $H$ with $d_{H}(v)\equiv\delta(v)~(\bmod~2)$ for each $v\in V(H)$, using at most $k$ operations from $S$. We prove that 1. if $S=\{ea\}$ or $S=\{ea,ed\}$, then CDPE($S$) can be solved in polynomial time; 2. if $\{vd\} \subseteq S\subseteq \{ea,ed,vd\}$, then CDPE($S$) is NP-complete and W[1]-hard when parameterized by $k$, even if $\delta\equiv 0$. Together with known results by Cai and Yang and by Cygan, Marx, Pilipczuk, Pilipczuk and Schlotter, our results completely classify the classical and parameterized complexity of the CDPE($S$) problem for all $S\subseteq \{ea,ed,vd\}$. We obtain the same classification for a natural variant of the CDPE($S$) problem on directed graphs, where the target is a weakly connected digraph in which the difference between the in- and out-degree of every vertex equals a prescribed value. As an important implication of our results, we obtain polynomial-time algorithms for the Eulerian Editing problem and its directed variant.Comment: 33 pages. An extended abstract of this paper will appear in the proceedings of FSTTCS 201