On retracts, absolute retracts, and folds in cographs

Let G and H be two cographs. We show that the problem to determine whether H is a retract of G is NP-complete. We show that this problem is fixed-parameter tractable when parameterized by the size of H. When restricted to the class of threshold graphs or to the class of trivially perfect graphs, the problem becomes tractable in polynomial time. The problem is also soluble when one cograph is given as an induced subgraph of the other. We characterize absolute retracts of cographs.Comment: 15 page

The complexity of Boolean surjective general-valued CSPs

Valued constraint satisfaction problems (VCSPs) are discrete optimisation problems with a $(\mathbb{Q}\cup\{\infty\})$-valued objective function given as a sum of fixed-arity functions. In Boolean surjective VCSPs, variables take on labels from $D=\{0,1\}$ and an optimal assignment is required to use both labels from $D$. Examples include the classical global Min-Cut problem in graphs and the Minimum Distance problem studied in coding theory. We establish a dichotomy theorem and thus give a complete complexity classification of Boolean surjective VCSPs with respect to exact solvability. Our work generalises the dichotomy for $\{0,\infty\}$-valued constraint languages (corresponding to surjective decision CSPs) obtained by Creignou and H\'ebrard. For the maximisation problem of $\mathbb{Q}_{\geq 0}$-valued surjective VCSPs, we also establish a dichotomy theorem with respect to approximability. Unlike in the case of Boolean surjective (decision) CSPs, there appears a novel tractable class of languages that is trivial in the non-surjective setting. This newly discovered tractable class has an interesting mathematical structure related to downsets and upsets. Our main contribution is identifying this class and proving that it lies on the borderline of tractability. A crucial part of our proof is a polynomial-time algorithm for enumerating all near-optimal solutions to a generalised Min-Cut problem, which might be of independent interest.Comment: v5: small corrections and improved presentatio

The Complexity of Approximately Counting Retractions

Let $G$ be a graph that contains an induced subgraph $H$. A retraction from $G$ to $H$ is a homomorphism from $G$ to $H$ that is the identity function on $H$. Retractions are very well-studied: Given $H$, the complexity of deciding whether there is a retraction from an input graph $G$ to $H$ is completely classified, in the sense that it is known for which $H$ this problem is tractable (assuming $\mathrm{P}\neq \mathrm{NP}$). Similarly, the complexity of (exactly) counting retractions from $G$ to $H$ is classified (assuming $\mathrm{FP}\neq \#\mathrm{P}$). However, almost nothing is known about approximately counting retractions. Our first contribution is to give a complete trichotomy for approximately counting retractions to graphs of girth at least $5$. Our second contribution is to locate the retraction counting problem for each $H$ in the complexity landscape of related approximate counting problems. Interestingly, our results are in contrast to the situation in the exact counting context. We show that the problem of approximately counting retractions is separated both from the problem of approximately counting homomorphisms and from the problem of approximately counting list homomorphisms --- whereas for exact counting all three of these problems are interreducible. We also show that the number of retractions is at least as hard to approximate as both the number of surjective homomorphisms and the number of compactions. In contrast, exactly counting compactions is the hardest of all of these exact counting problems

Graph Algorithms and Complexity Aspects on Special Graph Classes

Graphs are a very flexible tool within mathematics, as such, numerous problems can be solved by formulating them as an instance of a graph. As a result, however, some of the structures found in real world problems may be lost in a more general graph. An example of this is the 4-Colouring problem which, as a graph problem, is NP-complete. However, when a map is converted into a graph, we observe that this graph has structural properties, namely being (K_5, K_{3,3})-minor-free which can be exploited and as such there exist algorithms which can find 4-colourings of maps in polynomial time. This thesis looks at problems which are NP-complete in general and determines the complexity of the problem when various restrictions are placed on the input, both for the purpose of finding tractable solutions for inputs which have certain structures, and to increase our understanding of the point at which a problem becomes NP-complete. This thesis looks at four problems over four chapters, the first being Parallel Knock-Out. This chapter will show that Parallel Knock-Out can be solved in O(n+m) time on P_4-free graphs, also known as cographs, however, remains hard on split graphs, a subclass of P_5-free graphs. From this a dichotomy is shown on $P_k$-free graphs for any fixed integer $k$. The second chapter looks at Minimal Disconnected Cut. Along with some smaller results, the main result in this chapter is another dichotomy theorem which states that Minimal Disconnected Cut is polynomial time solvable for 3-connected planar graphs but NP-hard for 2-connected planar graphs. The third chapter looks at Square Root. Whilst a number of results were found, the work in this thesis focuses on the Square Root problem when restricted to some classes of graphs with low clique number. The final chapter looks at Surjective H-Colouring. This chapter shows that Surjective H-Colouring is NP-complete, for any fixed, non-loop connected graph H with two reflexive vertices and for any fixed graph H’ which can be obtained from H by replacing vertices with true twins. This result enabled us to determine the complexity of Surjective H-Colouring on all fixed graphs H of size at most 4

Finding vertex-surjective graph homomorphisms

The Surjective Homomorphism problem is to test whether a given graph G called the guest graph allows a vertex-surjective homomorphism to some other given graph H called the host graph. The bijective and injective homomorphism problems can be formulated in terms of spanning subgraphs and subgraphs, and as such their computational complexity has been extensively studied. What about the surjective variant? Because this problem is NP-complete in general, we restrict the guest and the host graph to belong to graph classes G and H, respectively. We determine to what extent a certain choice of G and H influences its computational complexity. We observe that the problem is polynomial-time solvable if H is the class of paths, whereas it is NP-complete if G is the class of paths. Moreover, we show that the problem is even NP-complete on many other elementary graph classes, namely linear forests, unions of complete graphs, cographs, proper interval graphs, split graphs and trees of pathwidth at most 2. In contrast, we prove that the problem is fixed-parameter tractable in k if G is the class of trees and H is the class of trees with at most k leaves, or if G and H are equal to the class of graphs with vertex cover number at most k

Finding vertex-surjective graph homomorphisms

The Surjective Homomorphism problem is to test whether a given graph G called the guest graph allows a vertex-surjective homomorphism to some other given graph H called the host graph. The bijective and injective homomorphism problems can be formulated in terms of spanning subgraphs and subgraphs, and as such their computational complexity has been extensively studied. What about the surjective variant? Because this problem is NP-complete in general, we restrict the guest and the host graph to belong to graph classes G and H , respectively. We determine to what extent a certain choice of G and H influences its computational complexity. We observe that the problem is polynomial-time solvable if H is the class of paths, whereas it is NP-complete if G is the class of paths. Moreover, we show that the problem is even NP-complete on many other elementary graph classes, namely linear forests, unions of complete graphs, cographs, proper interval graphs, split graphs and trees of pathwidth at most 2. In contrast, we prove that the problem is fixed-parameter tractable in k if G is the class of trees and H is the class of trees with at most k leaves, or if G and H are equal to the class of graphs with vertex cover number at most k

Finding vertex-surjective graph homomorphisms

The Surjective Homomorphism problem is to test whether a given graph G called the guest graph allows a vertex-surjective homomorphism to some other given graph H called the host graph. The bijective and injective homomorphism problems can be formulated in terms of spanning subgraphs and subgraphs, and as such their computational complexity has been extensively studied. What about the surjective variant? Because this problem is NP-complete in general, we restrict the guest and the host graph to belong to graph classes G and H , respectively. We determine to what extent a certain choice of G and H influences its computational complexity. We observe that the problem is polynomial-time solvable if H is the class of paths, whereas it is NP-complete if G is the class of paths. Moreover, we show that the problem is even NP-complete on many other elementary graph classes, namely linear forests, unions of complete graphs, cographs, proper interval graphs, split graphs and trees of pathwidth at most 2. In contrast, we prove that the problem is fixed-parameter tractable in k if G is the class of trees and H is the class of trees with at most k leaves, or if G and H are equal to the class of graphs with vertex cover number at most k