69 research outputs found
Parameterized complexity of edge-coloured and signed graph homomorphism problems
We study the complexity of graph modification problems for homomorphism-based
properties of edge-coloured graphs. A homomorphism from an edge-coloured graph
to an edge-coloured graph is a vertex-mapping from to that
preserves adjacencies and edge-colours. We consider the property of having a
homomorphism to a fixed edge-coloured graph . Given an edge-coloured graph
, can we perform graph operations so that the resulting graph has a
homomorphism to ? The operations we consider are vertex-deletion,
edge-deletion and switching (an operation that permutes the colours of the
edges incident to a given vertex). Switching plays an important role in the
theory of signed graphs, that are -edge-coloured graphs whose colours are
and . We denote the corresponding problems (parameterized by ) by
VERTEX DELETION -COLOURING, EDGE DELETION -COLOURING and SWITCHING
-COLOURING. These generalise -COLOURING (where one has to decide if an
input graph admits a homomorphism to ). Our main focus is when has order
at most , a case that includes standard problems such as VERTEX COVER, ODD
CYCLE TRANSVERSAL and EDGE BIPARTIZATION. For such a graph , we give a
P/NP-complete complexity dichotomy for all three studied problems. Then, we
address their parameterized complexity. We show that all VERTEX DELETION
-COLOURING and EDGE DELETION -COLOURING problems for such are FPT.
This is in contrast with the fact that already for some of order~,
unless P=NP, none of the three considered problems is in XP. We show that the
situation is different for SWITCHING -COLOURING: there are three
-edge-coloured graphs of order for which this is W-hard, and
assuming the ETH, admits no algorithm in time for inputs of size
. For the other cases, SWITCHING -COLOURING is FPT.Comment: 18 pages, 8 figures, 1 table. To appear in proceedings of IPEC 201
Graph modification for edge-coloured and signed graph homomorphism problems: parameterized and classical complexity
We study the complexity of graph modification problems with respect to
homomorphism-based colouring properties of edge-coloured graphs. A homomorphism
from edge-coloured graph to edge-coloured graph is a vertex-mapping
from to that preserves adjacencies and edge-colours. We consider the
property of having a homomorphism to a fixed edge-coloured graph . The
question we are interested in is: given an edge-coloured graph , can we
perform graph operations so that the resulting graph admits a homomorphism
to ? The operations we consider are vertex-deletion, edge-deletion and
switching (an operation that permutes the colours of the edges incident to a
given vertex). Switching plays an important role in the theory of signed
graphs, that are 2-edge-coloured graphs whose colours are the signs and
. We denote the corresponding problems (parameterized by ) by
VD--COLOURING, ED--COLOURING and SW--COLOURING. These problems
generalise -COLOURING (to decide if an input graph admits a homomorphism to
a fixed target ).
Our main focus is when is an edge-coloured graph with at most two
vertices, a case that is already interesting as it includes problems such as
VERTEX COVER, ODD CYCLE RANSVERSAL and EDGE BIPARTIZATION. For such a graph
, we give a P/NP-c complexity dichotomy for VD--COLOURING,
ED--COLOURING and SW--COLOURING. We then address their parameterized
complexity. We show that VD--COLOURING and ED--COLOURING for all such
are FPT. In contrast, already for some of order 3, unless P=NP, none of the
three problems is in XP, since 3-COLOURING is NP-c. We show that
SW--COLOURING is different: there are three 2-edge-coloured graphs of
order 2 for which SW--COLOURING is W-hard, and assuming the ETH, admits no
algorithm in time . For the other cases, SW--COLOURING is FPT.Comment: 17 pages, 9 figures, 2 table
Parameterized Complexity of Edge-Coloured and Signed Graph Homomorphism Problems
We study the complexity of graph modification problems with respect to homomorphism-based colouring properties of edge-coloured graphs. A homomorphism from an edge-coloured graph G to an edge-coloured graph H is a vertex-mapping from G to H that preserves adjacencies and edge-colours. We consider the property of having a homomorphism to a fixed edge-coloured graph H, which generalises the classic vertex-colourability property. The question we are interested in is the following: given an edge-coloured graph G, can we perform k graph operations so that the resulting graph admits a homomorphism to H? The operations we consider are vertex-deletion, edge-deletion and switching (an operation that permutes the colours of the edges incident to a given vertex). Switching plays an important role in the theory of signed graphs, that are 2-edge-coloured graphs whose colours are the signs + and -. We denote the corresponding problems (parameterized by k) by Vertex Deletion-H-Colouring, Edge Deletion-H-Colouring and Switching-H-Colouring. These problems generalise the extensively studied H-Colouring problem (where one has to decide if an input graph admits a homomorphism to a fixed target H). For 2-edge-coloured H, it is known that H-Colouring already captures the complexity of all fixed-target Constraint Satisfaction Problems.
Our main focus is on the case where H is an edge-coloured graph of order at most 2, a case that is already interesting since it includes standard problems such as Vertex Cover, Odd Cycle Transversal and Edge Bipartization. For such a graph H, we give a PTime/NP-complete complexity dichotomy for all three Vertex Deletion-H-Colouring, Edge Deletion-H-Colouring and Switching-H-Colouring problems. Then, we address their parameterized complexity. We show that all Vertex Deletion-H-Colouring and Edge Deletion-H-Colouring problems for such H are FPT. This is in contrast with the fact that already for some H of order 3, unless PTime = NP, none of the three considered problems is in XP, since 3-Colouring is NP-complete. We show that the situation is different for Switching-H-Colouring: there are three 2-edge-coloured graphs H of order 2 for which Switching-H-Colouring is W[1]-hard, and assuming the ETH, admits no algorithm in time f(k)n^{o(k)} for inputs of size n and for any computable function f. For the other cases, Switching-H-Colouring is FPT
Parameterised Counting in Logspace
Logarithmic space bounded complexity classes such as L and NL play a central role in space bounded computation. The study of counting versions of these complexity classes have lead to several interesting insights into the structure of computational problems such as computing the determinant and counting paths in directed acyclic graphs. Though parameterised complexity theory was initiated roughly three decades ago by Downey and Fellows, a satisfactory study of parameterised logarithmic space bounded computation was developed only in the last decade by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015).
In this paper, we introduce a new framework for parameterised counting in logspace, inspired by the parameterised space bounded models developed by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015). They defined the operators para_W and para_? for parameterised space complexity classes by allowing bounded nondeterminism with multiple-read and read-once access, respectively. Using these operators, they characterised the parameterised complexity of natural problems on graphs. In the spirit of the operators para_W and para_? by Stockhusen and Tantau, we introduce variants based on tail-nondeterminism, para_{W[1]} and para_{?tail}. Then, we consider counting versions of all four operators applied to logspace and obtain several natural complete problems for the resulting classes: counting of paths in digraphs, counting first-order models for formulas, and counting graph homomorphisms. Furthermore, we show that the complexity of a parameterised variant of the determinant function for (0,1)-matrices is #para_{?tail} L-hard and can be written as the difference of two functions in #para_{?tail} L. These problems exhibit the richness of the introduced counting classes. Our results further indicate interesting structural characteristics of these classes. For example, we show that the closure of #para_{?tail} L under parameterised logspace parsimonious reductions coincides with #para_? L, that is, modulo parameterised reductions, tail-nondeterminism with read-once access is the same as read-once nondeterminism.
Initiating the study of closure properties of these parameterised logspace counting classes, we show that all introduced classes are closed under addition and multiplication, and those without tail-nondeterminism are closed under parameterised logspace parsimonious reductions.
Also, we show that the counting classes defined can naturally be characterised by parameterised variants of classes based on branching programs in analogy to the classical counting classes.
Finally, we underline the significance of this topic by providing a promising outlook showing several open problems and options for further directions of research
Parameterised Counting in Logspace
Logarithmic space-bounded complexity classes such as L and NL play a central role in space-bounded computation. The study of counting versions of these complexity classes have lead to several interesting insights into the structure of computational problems such as computing the determinant and counting paths in directed acyclic graphs. Though parameterised complexity theory was initiated roughly three decades ago by Downey and Fellows, a satisfactory study of parameterised logarithmic space-bounded computation was developed only in the last decade by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015). In this paper, we introduce a new framework for parameterised counting in logspace, inspired by the parameterised space-bounded models developed by Elberfeld, Stockhusen and Tantau. They defined the operators paraW and paraβ for parameterised space complexity classes by allowing bounded nondeterminism with multiple-read and read-once access, respectively. Using these operators, they characterised the parameterised complexity of natural problems on graphs. In the spirit of the operators paraW and paraβ by Stockhusen and Tantau, we introduce variants based on tail-nondeterminism, paraW[1] and paraβtail. Then, we consider counting versions of all four operators and apply them to the class L. We obtain several natural complete problems for the resulting classes: counting of paths in digraphs, counting first-order models for formulas, and counting graph homomorphisms. Furthermore, we show that the complexity of a parameterised variant of the determinant function for (0, 1)-matrices is # paraβtailL-hard and can be written as the difference of two functions in # paraβtailL. These problems exhibit the richness of the introduced counting classes. Our results further indicate interesting structural characteristics of these classes. For example, we show that the closure of # paraβtailL under parameterised logspace parsimonious reductions coincides with # paraβL. In other words, in the setting of read-once access to nondeterministic bits, tail-nondeterminism coincides with unbounded nondeterminism modulo parameterised reductions. Initiating the study of closure properties of these parameterised logspace counting classes, we show that all introduced classes are closed under addition and multiplication, and those without tail-nondeterminism are closed under parameterised logspace parsimonious reductions. Finally, we want to emphasise the significance of this topic by providing a promising outlook highlighting several open problems and directions for further research
Graph Theory
Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
Topics in graph colouring and extremal graph theory
In this thesis we consider three problems related to colourings of graphs and one problem in extremal graph theory. Let be a connected graph with vertices and maximum degree . Let denote the graph with vertex set all proper -colourings of and two -colourings are joined by an edge if they differ on the colour of exactly one vertex.
Our first main result states that has a unique non-trivial component with diameter . This result can be viewed as a reconfigurations analogue of Brooks' Theorem and completes the study of reconfigurations of colourings of graphs with bounded maximum degree.
A Kempe change is the operation of swapping some colours , of a component of the subgraph induced by vertices with colour or . Two colourings are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes. Our second main result states that all -colourings of a graph are Kempe equivalent unless is the complete graph or the triangular prism. This settles a conjecture of Mohar (2007).
Motivated by finding an algorithmic version of a structure theorem for bull-free graphs due to Chudnovsky (2012), we consider the computational complexity of deciding if the vertices of a graph can be partitioned into two parts such that one part is triangle-free and the other part is a collection of complete graphs. We show that this problem is NP-complete when restricted to five classes of graphs (including bull-free graphs) while polynomial-time solvable for the class of cographs.
Finally we consider a graph-theoretic version formulated by Holroyd, Spencer and Talbot (2007) of the famous Erd\H{o}s-Ko-Rado Theorem in extremal combinatorics and obtain some results for the class of trees
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