That is, the problems arise in the FPT with respect to the tree width parameter of the graph [25]. The classic coloring problem is to color the vertices of the graph such that no pair of adjacent vertices has the same color.
Previous Work
Our Results
H-Free Coloring Problems
Related Work
The bounded degree bifurcation problem requires dividing the vertices of G into two groups A and B such that the maximum degrees in the induced subgraphs G[A] and G[B] are at most a and b respectively. Wu, Yuan, and Zhao [76] showed the NP-completeness of the variant that requires splitting the vertices of a graph G into two groups so that both induced graphs are acyclic.
Our Results
Karpi´nski [49] recently studied a problem that asks to color the vertices of the graph using 2 colors so that there is no monochromatic cycle of a fixed length. Other variants that place restrictions on the degree of the vertices within the partitions have also been studied [8, 24].
Happy Coloring Problems
Previous Work
In addition, the authors presented approximation algorithms with approximation ratios 1/2 and max{1/k,Ω(∆−3)} for k-MHE and k-MHV, respectively, where ∆ is the maximum degree of the graph. There are known (parameterized) algorithms for the Multiway Cut problem with the cut size as a parameter.
Our Results
Using the result from [26], we see that, for an arbitrary k, the k-MHE problem is NP-hard for planar graphs. Using the result from [27], we conclude that, when the number of precolored vertices is bounded, the k-MHE problem can be solved in linear time for graphs with bounded branch width.
Replacement Paths Problems
Previous Work
They used linear time algorithms for Range Minima Query (RMQ) [9] and integer sorting in their solution. 1In the referenced papers, the authors ignore the term TSP T(G), since they assume that either the shortest path trees are given or the constraint on the class of input graph for which linear time algorithms are known for SPT.
Our Results
Clique A subset of vertices C ⊆V(G) such that all pairs of vertices in C are adjacent is called a clique. Complete graph A graph in which all pairs of nodes are adjacent is called a complete graph.
Parameterized Complexity
Minor An undirected graph H is called a minor of G if H can be formed from G by deleting edges and vertices and contracting edges. Directed acyclic graph A directed graph without any directed cycle is called a directed acyclic graph (DAG).
Tree-Width
Branch-width
MSOL and Courcelle’s Theorem
Assume that ϕ is an MSOL formula and G is a graph equipped with the evaluation of all free variables ϕ. Here ||ϕ|| is the length of the MSOL formula and n is the number of points of the graph G.
Neighborhood Diversity
Problem Definitions
Question) Is there a coloring ˜c:V(G)→[k] such that ˜c|S = cand the number of happy edges is at least. Question) Is there a coloring ˜c:V(G)→[k] such that ˜c|S = cand the number of happy vertices is at least. Maximum weighted happy edges (weighted MHE). Production)A coloring ˜c: V(G)→[k] such that c|˜S = c maximizing the total weight of happy edges.
The decision versions of the weighted variants of happy color problems are defined as follows:.
Other Notations
Example) An undirected graph G and sets of vertices S1, S2,. Output) A set of edgesC ⊆E(G) of minimal cardinality, the removal of which disconnects every pair of vertices in every setSi.
Organization
The MSOL formulation, together with Courcell's theorem, implies linear time solvability on graphs of bounded tree width. Where||ϕ|| is the length of the MSOL formula and t is the tree width of the graph. In this thesis, we answer the above question by giving a 2O(t)·n time-explicit combinatorial algorithm for the cut matching problem, where t is the tree width of the graph.
We also show that the Matching Cut problem is tractable for graphs with limited neighborhood diversity and other structural parameters.
Graphs with Bounded Tree-width
Now we explain how to calculate Mi[Ψ] for each partition Ψ on the nodes of the nice tree decomposition. If r is the root of the nice tree decomposition, the graph G has a fitting cut if ∆r = +1. By induction and the correctness of Mi[Ψ] values we can conclude the correctness of the algorithm.
There is an algorithm with running timeO∗(2O(t)) that solves the Matching Cut problem, where t is the width of the graph tree.
Graphs with Bounded Neighborhood Diversity
If Pi is a set of size ≥3 and has only one adjacent set of size 1, then G has a matching cut. If Pi is an I-set of size 2 and is adjacent to an I-set of size 2 and a set of size 1, then the vertices in all these sets are given the same label. If Pi is an I-set of size 2 and is adjacent to only one I-set of size 2, then in these two sets each vertex is given a different label.
IfPi is an anI array of size 2 and is adjacent to two arrays of size 1, in these three arrays, each vertex will receive a different label.
Other Structural Parameters
The vertices of each label must be entirely in the same set of the matching cut. There is an O∗(22d) running time algorithm that solves the matching cut problem, where d is the neighborhood diversity of the graph. There is an O∗(2|S|) running time algorithm to solve the matching cut problem, where S is the twin cover of the graph.
The Chromatic Number of the graph is the minimum number of colors required to properly color the graph and is denoted by χ(G).
H-Free Coloring
K r -Free 2 -Coloring
The correctness of the algorithm derives from the correctness of the Mi[Ψ] values, which can be proven using bottom-up induction on the beautiful tree decomposition. G has a valid bipartition if there exists a Ψ such that Mr[Ψ] = 1, where is the root node of the nice decomposition of the tree.
H -Free 2 -Coloring
ΓAi is the set of sequences that can become H in the future at an ancestral (insert/join) node of the tree decomposition. Now we explain how to calculate Mi[Ψ] values at the leaf, introduce, forget and merge nodes of the beautiful tree decomposition. The correctness of the algorithm is implied by the correctness of Mi[Ψ] values, which can be proven using a bottom-up induction on the nice tree decomposition.
The time complexity at each of the nodes in the tree decomposition is as follows: constant time at leaf nodes, O(2O(tr)) time at insert nodes, O(2O(tr)) time at forget nodes and O(2O(tr) )) time at junction nodes.
H -Free q -Coloring
Note that if is a legal sequence at node j with respect to Az, then RepFG(s, v) is also a legal sequence at node i with respect to Az. Join node: Let i be a join node, j1, j2 are respectively the left and right children of node i. Our techniques can also be used to compute the H-free chromatic number of the graph by searching for the smallest q for which there is a H-free q-staining.
There is an O(tO(tr)·nlogt) time algorithm to compute the H-Free Chromatic Number on the graph whose tree width is at most t.
H-(Subgraph)Free Coloring
- C 4 -(Subgraph)Free 2 -Coloring
- C r -(Subgraph)Free 2 -Coloring
- H -(Subgraph)Free 2 -Coloring
- H -(Subgraph)Free q -Coloring
- Computing T v [i, H ]
- Computing T v [i, U ]
- Generating all optimal happy vertex colorings
Suppose there exists a 4-tuple Ψ such that Mr[Ψ] = 1, where r is the root of the neat tree decomposition. If there is such a partition (A, B), then G[Ai] and G[Bi] can have at most partial paths of length r−2, which can lead to a cycle of length r at an ancestor node j (insert node or join node) of the neat tree decomposition (see figures 4.4 and 4.5). G has a valid bipartition if there exists a 4-tuple Ψ such that Mr[Ψ] = 1, where r is the root of the neat tree decomposition.
There is an O(tO(tr) · nlogt) time algorithm to calculate the H-(Subgraph) free chromatic number of the graph whose tree width is at most t.
Algorithm for k -MHE problem for Trees
Generating all optimal happy edge colorings
If at a vertex v the color(parent(v)) is present in L(v), then we assign the color(parent(v)) to v in the optimal coloring. Here we point out that this scheme may miss some optimal colorations if color(parent(v)) is not present in L(v) but is present in the set of colors with a frequency one lower than the maximum frequency. In this case, we can assign the color(parent(v)) to v even though the color(parent(v)) is not present in L(v).
Although color(parent(v)) is not present in L(v), we can assign color(parent(v)) to tov since it has zero frequency at v.
MHV and MHE on Complete Graphs
For each node v ∈C, regardless of the color we give to v, we can find at most p edges between the edges from the nodes in S to v. In fact, we can achieve exactly p· |C|+|E( C)| happy edges by assigning one color to all vertices in C. Specifically, we color all uncolored vertices with the color used p times, completing the proof.
Indeed, the procedure described in the proof can be used as long as every previously painted vertex is adjacent to every unpainted vertex.
Hardness Results for Happy Coloring Problems
Multiway Cutis NP-hard for planar graphs [26] when k, the number of terminals, is not fixed. The algorithm runs in linear time when the branch width and the number of node sets are fixed. When the branch width of the graph and the number of pre-colored nodes are bounded, there is a linear time algorithm for the k-MHE problem.
Thus, when both the number of precolored vertices and the branch width are constant, the k-MHE problem can be solved in linear time.
A Linear Kernel for Weighted MHE
Polynomial Time Algorithm for Subproblems of Weighted MHE 72
Tv[i] : maximum total weight of happy edges touching subtree Tv when vertex v is colored i. Tv[ı] : Maximum total weight of happy edges touching subtree Tv when vertex v is colored with a color other than i, ie. If Wp is the total weight of lucky edges in the initial partial coloring, Wp +Tr[∗].
Input: A weighted undirected graph G with S⊆V(G) pre-colored nodes under partial node coloring c:S→[k],V(G)\S induces a tree and node r∈V(G)\S as the root of the tree.
Tree-Width
So in any component of H, the number of vertices is at most the number of edges. Furthermore, after we have bounded the number of precolored and uncolored vertices, the claimed kernel follows. For the special case of k= 3, the weighted DMHE problem admits an algorithm that runs in O∗(1.89`) time.
Put differently, Gi is formed from Gj by adding v and a number of edges from v to vertices in Xj.
Neighborhood Diversity
Let X1 and X2 be the neighbors of nodes in Q1 and Q2 in color classes C1 and C2, respectively, as shown in Figure 5.2. For every k ≥ 1, the MHE can be solved in O*(2d) time, where d is the neighborhood diversity of the input graph. For each k ≥ 1, MHV can be solved in O*(2d) time, where d is the neighborhood diversity of the input graph.
In the remainder of the section, we describe the algorithm for calculating RSP using Ts and Tt (phase (ii)).
Labeling the nodes of G
For the Matching Cut problem, we introduced an O∗(2O(t)) time algorithm, where t is the width of the graph tree. It is interesting to study the complexity of the Matching Cut problem parameterized by the size of the cut. The Matching Cut problem parameterized by cut size is equivalent to the problem of deciding whether the line graph of G has at most independent vertex cuts of size.
The complexity of the MHE and MHV problems with respect to the tree width parameter alone can be investigated.
An example graph
Tree decomposition of the graph G
A nice tree decomposition of G
An example graph H
A cycle of length 20 formed using the paths from left and right subtrees at
Potential replacement paths for the edge e i . The zig-zag lines represent a
