Graph Editing to a Given Neighbourhood Degree List is Fixed-Parameter Tractable

Graph editing problems have a long history and have been widely studied, with applications in biochemistry and complex network analysis. They generally ask whether an input graph can be modified by inserting and deleting vertices and edges to a graph with the desired property. We consider the problem \textsc{Graph-Edit-to-NDL} (GEN) where the goal is to modify to a graph with a given neighbourhood degree list (NDL). The NDL lists the degrees of the neighbours of vertices in a graph, and is a stronger invariant than the degree sequence, which lists the degrees of vertices. We show \textsc{Graph-Edit-to-NDL} is NP-complete and study its parameterized complexity. In parameterized complexity, a problem is said to be fixed-parameter tractable with respect to a parameter if it has a solution whose running time is a function that is polynomial in the input size but possibly superpolynomial in the parameter. Golovach and Mertzios [ICSSR, 2016] studied editing to a graph with a given degree sequence and showed the problem is fixed-parameter tractable when parameterized by $\Delta+\ell$, where $\Delta$ is the maximum degree of the input graph and $\ell$ is the number of edits. We prove \textsc{Graph-Edit-to-NDL} is fixed-parameter tractable when parameterized by $\Delta+\ell$. Furthermore, we consider a harder problem \textsc{Constrained-Graph-Edit-to-NDL} (CGEN) that imposes constraints on the NDLs of intermediate graphs produced in the sequence. We adapt our FPT algorithm for \textsc{Graph-Edit-to-NDL} to solve \textsc{Constrained-Graph-Edit-to-NDL}, which proves \textsc{Constrained-Graph-Edit-to-NDL} is also fixed-parameter tractable when parameterized by $\Delta+\ell$. Our results imply that, for graph properties that can be expressed as properties of NDLs, editing to a graph with such a property is fixed-parameter tractable when parameterized by $\Delta+\ell$. We show that this family of graph properties includes some well-known graph measures used in complex network analysis

Graph Editing to a Given Neighbourhood Degree List is Fixed-Parameter Tractable

Graph editing problems have a long history and have been widely studied, with applications in biochemistry and complex network analysis. They generally ask whether an input graph can be modified by inserting and deleting vertices and edges to a graph with the desired property. We consider the problem \textsc{Graph-Edit-to-NDL} (GEN) where the goal is to modify to a graph with a given neighbourhood degree list (NDL). The NDL lists the degrees of the neighbours of vertices in a graph, and is a stronger invariant than the degree sequence, which lists the degrees of vertices. We show \textsc{Graph-Edit-to-NDL} is NP-complete and study its parameterized complexity. In parameterized complexity, a problem is said to be fixed-parameter tractable with respect to a parameter if it has a solution whose running time is a function that is polynomial in the input size but possibly superpolynomial in the parameter. Golovach and Mertzios [ICSSR, 2016] studied editing to a graph with a given degree sequence and showed the problem is fixed-parameter tractable when parameterized by $\Delta+\ell$, where $\Delta$ is the maximum degree of the input graph and $\ell$ is the number of edits. We prove \textsc{Graph-Edit-to-NDL} is fixed-parameter tractable when parameterized by $\Delta+\ell$. Furthermore, we consider a harder problem \textsc{Constrained-Graph-Edit-to-NDL} (CGEN) that imposes constraints on the NDLs of intermediate graphs produced in the sequence. We adapt our FPT algorithm for \textsc{Graph-Edit-to-NDL} to solve \textsc{Constrained-Graph-Edit-to-NDL}, which proves \textsc{Constrained-Graph-Edit-to-NDL} is also fixed-parameter tractable when parameterized by $\Delta+\ell$. Our results imply that, for graph properties that can be expressed as properties of NDLs, editing to a graph with such a property is fixed-parameter tractable when parameterized by $\Delta+\ell$. We show that this family of graph properties includes some well-known graph measures used in complex network analysis

The Parameterized Complexity of Degree Constrained Editing Problems

This thesis examines degree constrained editing problems within the framework of parameterized complexity. A degree constrained editing problem takes as input a graph and a set of constraints and asks whether the graph can be altered in at most k editing steps such that the degrees of the remaining vertices are within the given constraints. Parameterized complexity gives a framework for examining problems that are traditionally considered intractable and developing efficient exact algorithms for them, or showing that it is unlikely that they have such algorithms, by introducing an additional component to the input, the parameter, which gives additional information about the structure of the problem. If the problem has an algorithm that is exponential in the parameter, but polynomial, with constant degree, in the size of the input, then it is considered to be fixed-parameter tractable. Parameterized complexity also provides an intractability framework for identifying problems that are likely to not have such an algorithm. Degree constrained editing problems provide natural parameterizations in terms of the total cost k of vertex deletions, edge deletions and edge additions allowed, and the upper bound r on the degree of the vertices remaining after editing. We define a class of degree constrained editing problems, WDCE, which generalises several well know problems, such as Degree r Deletion, Cubic Subgraph, r-Regular Subgraph, f-Factor and General Factor. We show that in general if both k and r are part of the parameter, problems in the WDCE class are fixed-parameter tractable, and if parameterized by k or r alone, the problems are intractable in a parameterized sense. We further show cases of WDCE that have polynomial time kernelizations, and in particular when all the degree constraints are a single number and the editing operations include vertex deletion and edge deletion we show that there is a kernel with at most O(kr(k + r)) vertices. If we allow vertex deletion and edge addition, we show that despite remaining fixed-parameter tractable when parameterized by k and r together, the problems are unlikely to have polynomial sized kernelizations, or polynomial time kernelizations of a certain form, under certain complexity theoretic assumptions. We also examine a more general case where given an input graph the question is whether with at most k deletions the graph can be made r-degenerate. We show that in this case the problems are intractable, even when r is a constant

Fixed-Parameter Tractable Distances to Sparse Graph Classes

We show that for various classes $\mathcal{C}$ of sparse graphs, and several measures of distance to such classes (such as edit distance and elimination distance), the problem of determining the distance of a given graph $\small{G}$ to $\mathcal{C}$ is fixed-parameter tractable. The results are based on two general techniques. The first of these, building on recent work of Grohe et al. establishes that any class of graphs that is slicewise nowhere dense and slicewise first-order definable is FPT. The second shows that determining the elimination distance of a graph $\small{G}$ to a minor-closed class $\mathcal{C}$ is FPT. We demonstrate that several prior results (of Golovach, Moser and Thilikos and Mathieson) on the fixed-parameter tractability of distance measures are special cases of our first method

Graph editing problems with extended regularity constraints

© 2017 Graph editing problems offer an interesting perspective on sub- and supergraph identification problems for a large variety of target properties. They have also attracted significant attention in recent years, particularly in the area of parameterized complexity as the problems have rich parameter ecologies. In this paper we examine generalisations of the notion of editing a graph to obtain a regular subgraph. In particular we extend the notion of regularity to include two variants of edge-regularity along with the unifying constraint of strong regularity. We present a number of results, with the central observation that these problems retain the general complexity profile of their regularity-based inspiration: when the number of edits k and the maximum degree r are taken together as a combined parameter, the problems are tractable (i.e. in FPT), but are otherwise intractable. We also examine variants of the basic editing to obtain a regular subgraph problem from the perspective of parameterizing by the treewidth of the input graph. In this case the treewidth of the input graph essentially becomes a limiting parameter on the natural k+r parameterization

Modification to planarity is fixed parameter tractable

A replacement action is a function L that maps each k-vertex labeled graph to another k-vertex graph. We consider a general family of graph modification problems, called L-Replacement to C, where the input is a graph G and the question is whether it is possible to replace in G some k-vertex subgraph H of it by L(H) so that the new graph belongs to the graph class C. L-Replacement to C can simulate several modification operations such as edge addition, edge removal, edge editing, and diverse completion and superposition operations. In this paper, we prove that for any action L, if C is the class of planar graphs, there is an algorithm that solves L-Replacement to C in O(|G| 2 ) steps. We also present several applications of our approach to related problems.publishedVersio

Modification to Planarity is Fixed Parameter Tractable

A replacement action is a function L that maps each k-vertex labeled graph to another k-vertex graph. We consider a general family of graph modification problems, called L-Replacement to C, where the input is a graph G and the question is whether it is possible to replace in G some k-vertex subgraph H of it by L(H) so that the new graph belongs to the graph class C. L-Replacement to C can simulate several modification operations such as edge addition, edge removal, edge editing, and diverse completion and superposition operations. In this paper, we prove that for any action L, if C is the class of planar graphs, there is an algorithm that solves L-Replacement to C in O(|G|^{2}) steps. We also present several applications of our approach to related problems

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

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