On the fixed-parameter tractability of the maximum 2-edge-colorable subgraph problem
A -edge-coloring of a graph is an assignment of colors to edges of the graph such that adjacent edges receive different colors. In the maximum -edge-colorable subgraph problem we are given a graph and an integer , the goal is to find a -edge-colorable subgraph with maximum number of edges together with its -edge-coloring. In this paper, we consider the maximum 2-edge-colorable subgraph problem and present some results that deal with the fixed-parameter tractability of this problem.
keywords:Edge-coloring, maximum -edge-colorable subgraph, exact algorithm, fixed-parameter tractability
In this paper, we consider finite, undirected graphs without loops or multiple edges. The set of vertices and edges of a graph is denoted by and , respectively. denotes the degree of a vertex of . Let and be the minimum and maximum degree of vertices of . Let and be the radius and diameter of .
A matching in a graph is a subset of such that no vertex of is incident to two edges from it. A maximum matching is a matching that contains the largest possible number of edges.
For , a graph is -edge colorable, if its edges can be assigned colors from a set of colors so that adjacent edges receive different colors. The smallest , such that is -edge-colorable is called chromatic index of and is denoted by . The classical theorem of Shannon states that for any multi-graph Shannon:1949 ; stiebitz:2012 . Moreover, the classical theorem of Vizing states that for any multi-graph stiebitz:2012 ; vizing:1964 . Here denotes the maximum multiplicity of an edge of . A multi-graph is class I if , otherwise it is class II.
If , we cannot color all edges of with colors. Therefore, it is natural to investigate the maximum number of edges that one can color with colors. A subgraph of is called maximum -edge-colorable, if is -edge-colorable and contains maximum number of edges among all -edge-colorable subgraphs of . For and a graph let
Clearly, a -edge-colorable subgraph is maximum if it contains exactly edges. Observe that is the size of a maximum matching of . We will shorten this notation to .
There are many papers where the ratio has been investigated. bollobas:1978 ; henning:2007 ; nishizeki:1981 ; nishizeki:1979 ; weinstein:1974 prove lower bounds for this ratio in case of regular graphs and . For regular graphs of high girth the bounds are improved in flaxman:2007 . Albertson and Haas investigated the problem in haas:1996 ; haas:1997 when is a cubic graph. See also samvel:2010 , where it is shown that for every cubic graph and . Moreover, samvel:2014 proves that for any cubic graph , and in samvel:2010 ; corrigendum Mkrtchyan et al. showed that for any cubic graph . Finally, in LianaDAM:2019 , it is shown that the sequence is convex in the class of bipartite graphs.
Bridgeless cubic graphs that are not -edge-colorable are called snarks cavi:1998 , and the ratio for snarks is investigated by Steffen in steffen:1998 ; steffen:2004 . This lower bound has also been investigated in the case when the graphs need not be cubic in miXumbFranciaciq:2013 ; Kaminski:2014 ; Rizzi:2009 . Kosowski and Rizzi have investigated the problem from the algorithmic perspective Kosowski:2009 ; Rizzi:2009 . Since the problem of finding a -edge-colorable graph in an input graph is NP-complete for every fixed , it is natural to investigate the (polynomial) approximability of the problem. In Kosowski:2009 , for each an algorithm for the problem is presented. There for each fixed value of , algorithms are proved to have certain approximation ratios and these ratios are tending to as goes to infinity.
Some structural properties of maximum -edge-colorable subgraphs of graphs are proved in samvel:2014 ; MkSteffen:2012 . There it is shown that every set of disjoint cycles of a graph with can be extended to a maximum -edge colorable subgraph. Moreover, there it is shown that any maximum -edge colorable subgraph of a graph is always class I. Finally, if is a graph of girth (the length of the shortest cycle) and is a maximum -edge colorable subgraph of , then , The bound is best possible as there is an example attaining it.
In this paper, we deal with the exact solvability of the maximum -edge-colorable subgraph problem. Its precise formulation is the following:
(Maximum -edge-colorable subgraph) Given a graph and an integer , find a -edge-colorable subgraph with maximum number of edges together with its -edge-coloring.
We investigate this problem from the perspective of fixed-parameter tractability. Recall that an algorithmic problem is fixed-parameter tractable with respect to a parameter , if there is an exact algorithm solving , whose running-time is . Here is some (computable) function of , is the length of the input and is a polynomial. In kEdgeColoringFPT the -edge-coloring problem is considered, which is formulated as follows:
(-edge-coloring) Given a graph and an integer , check whether is -edge-colorable.
There it is shown that for each fixed , the -edge-coloring problem is fixed-parameter tractable with respect to the number of maximum degree vertices of the input graph. Observe that the maximum -edge-colorable subgraph problem is harder than -edge-coloring, as if we can construct a maximum -edge-colorable subgraph of the input graph , then in order to see that whether is -edge-colorable, we just need to check whether . If one considers the edge-coloring problem, where for an input graph , we need to find a -edge-coloring of , then in KowalikSIDMA:2018 it is stated that a major challenge in the area is to find an exact algorithm for this problem whose running-time is . Observe that the maximum -edge-colorable subgraph problem is harder than edge-coloring. If we are able to solve the maximum -edge-colorable subgraph problem in time , then we can solve the Edge-Coloring problem in time . In order to see this, just observe that we can do a binary search on , solve the maximum -edge-colorable problem and find an edge-coloring of with the smallest number of colors. Here we used the fact that any graph is -edge-colorable.
In this paper, we focus on the maximum 2-edge-colorable subgraph problem which is the restriction of the problem to the case . We present some results that deal with the fixed-parameter tractability of this problem with respect to various graph-theoretic parameters. The results obtained in this paper are summarized in Table 1. For the notions, facts and concepts that are not explained in the paper the reader is referred to FPTbook ; west:1996 .
|Theorem 5||, , , Number of maximum-degree vertices||NP-hard|
|Corollary 1||Maximum matching||Yes|
|Theorem 4||Dimension of the cycle space||Yes|
2 Some auxiliary results
In this section, we present some results that will be used in obtaining the main results of the paper. Below we assume that is the set of natural numbers.
(Sasak:2010 ) Let be an algorithmic problem, and let and be some parameters. Assume that there is a (computable) function , such that for any instance of , we have . Then if is FPT with respect to , then it is FPT with respect to .
In holyer:1981 , Holyer has shown that checking whether a cubic graph is 3-edge-colorable is an NP-complete problem. For a cubic graph , let be defined as:
This parameter is introduced and is investigated in steffen:2004 . In particular, their it is observed there that for any cubic graph . This means that can be zero or at least two, and the 3-edge-coloring problem in cubic graphs amounts to deciding which of these two cases holds. For our purposes we will consider the following restriction of 3-edge-coloring problem in cubic graphs:
Problem 1: For a fixed integer , consider a decision problem, whose input is a cubic graph , in which is from the set . The goal is to check whether is 3-edge-colorable, that is, whether .
For each fixed , Problem 1 is -complete.
The case when corresponds to the usual 3-edge-coloring problem in cubic graphs. Thus, we can assume that . We reduce the 3-edge-coloring problem of cubic graphs to this problem. Let be any cubic graph. Consider a cubic graph obtained from vertex disjoint copies of . Observe that , hence can be constructed from in linear time. Now, it is easy to see that is 3-edge-colorable if and only if is 3-edge-colorable. Moreover, . Hence, is either zero or at least . The proof is complete. ∎
For any cubic graph .
3 Main results
In this section, we present our main results about the maximum 2-edge-colorable subgraph problem. If is the number of edges of the input graph , then clearly we can generate all subgraphs/subsets of , and check each of them for -edge-colorability. In great contrast with -edge-colorability with , checking 2-edge-colorability can be done in polynomial time. A subgraph of is 2-edge-colorable if and only if it has maximum degree at most two, and it contains no component that is an odd cycle. Clearly this can be checked in polynomial time. The running time of this trivial, brute-force algorithm is . We will refer to this algorithm as trivial or brute-force algorithm.
The first parameter with respect to which we will investigate our problem is the radius of the graph.
If , then the maximum 2-edge-colorable subgraph problem cannot be parameterized with respect to the .
Assume the opposite, that is the problem is FPT with respect to the . Consider Problem 1 with . By Lemma 2 it is -complete. Let us take an arbitrary cubic graph with either zero or at least . Take a new vertex , who is joined to every vertex of . Let be the resulting graph.
Let us show that if and only if is 3-edge-colorable. Let be a 3-edge-colorable. Then it admits a pair of edge-disjoint perfect matchings. Hence, these perfect matchings form a 2-edge-colorable subgraph in . Thus, . Now, assume that is not 3-edge-colorable, hence . By Theorem 1:
since . Hence, if , then is 3-edge-colorable.
Now, if the problem is FPT with respect to , then since in graphs that we obtained from , we have ( is of distance one from any other vertex), we have that in polynomial time we can find a maximum 2-edge-colorable subgraph in . Thus, by the previous remark, we ca use this polynomial algorithm to decide whether , or equivalently, whether is 3-edge-colorable in the class of cubic graphs, in which is zero or at least . Since by Lemma 2, the latter problem is -complete, we have . The proof is complete. ∎
If , then the maximum 2-edge-colorable subgraph problem cannot be parameterized with respect to the .
In the following, we show that the maximum 2-edge-colorable subgraph problem is fixed-parameter tractable respect to the pathwidth of the graph. We first recall some basic definitions that we use in Theorem 3. Informally, a path decomposition of a graph is a way of representing as a path-like structure.
Definition 1 (fomin:2010 ).
A path decomposition of a graph is a set of subsets of , that is for each , called bags, such that
(i) for every there exists with ;
(ii) for every , there exists with ;
(iii) for every three bags , , and , with , it holds that .
The width of a path decomposition equals , and the pathwidth of a graph , is the minimum width of a path decomposition of . To avoid confusion between the vertices of the graph, and the ones in the path , we will call nodes the vertices in the path decomposition. A property of path decompositions fomin:2010 , called here pathwidth separator property, is that for every three nodes , , and , with , each path that connects a vertex in with a vertex in contains a vertex in . Thus, node separates the vertices in from the ones in . Our dynamic programming algorithm works on a particular type of path decomposition called nice, which can be always constructed in linear time from any path decomposition, maintaining the same width.
Definition 2 (fomin:2010 ).
A path decomposition of a graph is nice if , and for every there is a vertex , such that either (introduce node), or (forget node).
Since property (iii) in Definition 1 says that every vertex belongs to a consecutive set of bags, the number of nodes in a nice path decomposition is at most twice the number of vertices in . In Figure 2 there is a path decomposition, and in Figure 3 there is a nice path decomposition, both with width 2, and both for the graph in Figure 2.
The maximum 2-edge-colorable subgraph problem is FPT with respect to the pathwidth , and can be solved in time via a dynamic programming algorithm.
Let be a graph where is the set of vertices, is the set of edges, and with a path decomposition of width . Assume that is the set of colours where , and are true colours, while is dummy and means ’not coloured’. Let be a function, which is equal to 1 if and only if the input is a true colour, 0 otherwise (i.e., , , and ). If is a function, then is called the color assigned to the edge . In order to avoid cluttered notation, in the following we will write instead of .
The first step of the algorithm is to compute a nice path decomposition with the same width , which can be done in linear time fomin:2010 . Denote by the subgraph induced by the vertices in . Let be the maximum value of 2-edge-colorable subgraph problem on , where is a collection of subsets of colors , with , that satisfies the following constraints.
The colors incident to the vertex are those in , for every .
Here, as usual, a color is incident to a vertex if it is used at least in one edge incident to . Notice that only the dummy color can be incident more than once to a vertex, because it means that an edge is not colored.
At each node of , we compute the values , for every possible collection , via a dynamic programming algorithm that starts at , ends at , and exploits the pathwidth separator property. If there is no solution of the constrained problem, then we set . Since in there is only one vertex, and in there are no edges, one of the following conditions hold:
In fact, there cannot be true colors in , so there is only one solution with , and optimum value 0.
In any introduce node , the value , for a specific collection of color sets, is computed by solving the following maximisation problem.
where is the set of the vertices incident to . The fourth constraint makes explicit how the colours incident to are used on the edges , with . The first three constraints, instead, are used to properly define , that is a collection of color subsets for derived from . In particular, the first one states that is equal to for every vertex non adjacent to ; the second states that is equal to if the color used for the edge is ; and the third states that is equal to minus the true colour used for the edge , because it cannot be used again for the edges in incident to . The objective function sums the already computed optimum with the number of edges in incident to that receive a true color.
For any forget node , the value for a specific collection of incident color sets , with , is computed by solving the following maximisation problem.
In fact, the value is essentially the maximum value of for every possible subset of colors compatible with every , that means for every adjacent to .
Once this dynamic programming algorithm stops, the optimum of 2-edge-colorable subgraph problem is the maximum value , for every possible collection of color subsets of the incident colors in each vertex . Now, we compute the complexity of the algorithm. At each introduce node, we solve at most problems 1, once for every possible subset of incident colors , and for every that are at most . Moreover, for each specific in 1, the maximum number of edges with is , because . If , we can color these edges only if , and we can do it in at most different ways. In fact, if contains one true colour, there are possibilities to select the edge associated with it; while, if contains both the true colours, the possibilities are . Clearly, if , there can be less ways of colouring these edges. This means that, for a specific , Problem 1 has at most solutions, and the complexity for an introduce node is .
At any forget node, the complexity is at most , given by all the subsets of colors , with , and all the possible subsets of colors for the forget vertex . In conclusion, since there are at most introduce nodes and forget nodes, the time complexity of the dynamic programming algorithm is . ∎
The maximum 2-edge-colorable subgraph problem is FPT with respect to .
Let be the smallest number of edges of such that any vertex of is incident to at least one of these edges. By the classical Gallai theorem west:1996 , we have that if the graph has no isolated vertices, then
Since , we have
Thus, Corollary 1 and Lemma 1 imply that the maximum 2-edge-colorable subgraph problem is FPT with respect to . Observe that isolated vertices play no role in the maximum -edge-colorable subgraph problem, thus we can assume that the input graph contains none of them.
Also, observe that the parameterization with respect to can be interpreted as parameterization with respect to . One may wonder, whether we can strengthen this result, by showing that the maximum 2-edge-colorable subgraph problem is FPT with respect to ? The answer to this question is negative unless . If a cubic graph is 3-edge-colorable, then it must be bridgeless. Thus, by Holyer’s result the maximum 2-edge-colorable subgraph problem is -hard for bridgeless cubic graphs. By the classical Petersen theorem west:1996 , bridgeless cubic graphs have a perfect matching. Thus, in this class we have . Hence, if , the maximum 2-edge-colorable subgraph problem cannot be FPT with respect to .
One can consider the decision version of the maximum 2-edge-colorable subgraph problem, where for a given graph and an integer , one needs to check whether . It turns out that this problem is FPT with respect to . In order to see this, just observe that if in the input graph , then clearly , hence the instance is a “yes” instance. On the other hand, if , then the FPT algorithm with respect to (Corollary 1) will in fact will be an FPT algorithm with respect to (Lemma 1).
Below we will parameterize the maximum 2-edge-colorable subgraph problem with respect to the dimension of the cycle space of a graph. Recall that if is any graph with components then the dimension of its cycle space is given by the following formula:
For our parameterization, we will require the following lemma:
Let be a forest, and let be a set of vertices of . Assume that is a partition of . Then there is a linear time algorithm that finds a largest 2-edge-colorable subgraph such that the vertices of are not incident to edges with color ().
Clearly, we can assume that is a tree, otherwise we can find a largest 2-edge-colorable subgraph respecting constraints in each of the components, and by taking their union we will get an exact solution for the whole forest. Moreover, the constraints on the vertices in can be seen as: only color can be used on the edges incident to a vertex in ; and only color can be used on the edges incident to a vertex in . In the following, we say that a color is incident to a vertex, if it is used at least on one of the edges incident to the vertex. In order to describe the algorithm, we add the extra dummy color that means ’not colored’, so the set of available colors becomes , where , and are called true colors.
Let be a tree with vertices and edges. Assume that is an allocation of the available colors for each vertex in . Let be a profit function that is equal to 0, if the input is the dummy color , and is 1, otherwise. Let be the root of , and let be the subgraph induced by and all the descendants of in .
Now, we describe a dynamic programming algorithm that finds a largest 2-edge-colorable subgraph on , with these additional constraints:
the colors that can be incident to each vertex are the ones in .
We call this problem . Clearly, we can write the original problem of the lemma as with a specific color allocation, where for each vertex ; for each vertex ; and for each vertex .
In the algorithm, we compute for each vertex , and for every , which is equal to the optimum value of restricted to the subgraph , and with the additional following constraint:
the colors incident to are those in .
If there is no solution, we set . In particular, the algorithm starts from the leaves and goes up to the root , and the optimal value of is the maximum , for every .
If is a leaf, and a specific color subset of , one of the following conditions holds:
In fact, since there are no edges in when is a leaf, there cannot be any color incident to in .
For each internal vertex , we suppose , as it is always possible to use the dummy color . If is an internal vertex with sons , we compute for any subset by using the values for every ’s son . For every , denote by the set containing the vertices in the subgraph minus the vertices in each subgraph , with . Let be the subgraph induced by the vertices in . For every we compute , which equals the maximum value of restricted to the subgraph , and with the following additional constraint:
the colors incident to are those in .
If there is no solution, we set . Notice that is equivalent to . Now, we see how to compute for every , recursively.
If , there is only one edge incident to in , i,.e., . So, we can set , where contains only the color assigned to . We compute solving the following problem.
If , we calculate by using the values . We solve the following maximisation problem for every non empty subset and any colour for the edge , such that , and .
In fact, for a specific color , we get the best value for every that is compatible with . The compatibility is guaranteed by the constraint that does not allow to choose a subset (the problem’s variable) that contains if it is a true color.
If , we calculate by using the values . We solve the following maximisation problem for every non empty subset and any color for the edge , such that , and .
The idea is that, for every , and for every that are compatible, we search the subset (the problem’s variable), compatible with , which maximise the number of edges with true colors in , that is the objective function. In the objective function, refers to the subgraph , while is the value already computed.
The time complexity to compute for all the sons of an internal vertex is . In fact, for we solve Problem 3 for every , so at most times; at each of the steps, with , we solve Problem 4 at most for every subset , and for every color . Since there are less that non empty subsets , at most possible colors for , and at most subsets (the problems’ variable), the time complexity for computing for every ’s sons, is . Then, this is the time complexity to compute for an internal vertex with sons.
In conclusion, starting from the leaves, we can compute for every internal node , from the lowest level of the tree until we reach . Since we need time for each internal node with sons, the total running-time will be which is , as the number of edges in a tree is . ∎
We are ready to obtain the next result.
The maximum 2-edge-colorable subgraph problem is FPT with respect to the dimension of the cycle space.
First of all let us show that it suffices to parameterize the problem when is connected. Assume that are the connected components of . Let us run the FPT algorithm on each . Since the union of maximum 2-edge-colorable subgraphs in will give rise to a maximum 2-edge-colorable subgraph of , this algorithm will solve the problem exactly. Let us estimate its running-time. If is the monotone (computable) function for parameterizing in the class of connected graphs, for the running time we will have
Since for any
and and is monotone, we have that the total running-time is bounded by
Thus, it suffices to parameterize the problem when the input graph is connected. In this case, we need to parameterize the problem with respect to . Our parameterization will work for , hence it will be FPT with respect to (Lemma 1).
We consider two cases. If , then we have
Thus, the trivial brute-force algorithm will have a running time that is bounded in terms of .
Thus, we can assume that . Since is connected, it has a spanning tree . Observe that contains edges. Thus, there are at most edges of that are outside . Any maximum 2-edge-colorable subgraph of colors some edges outside . Thus, we can guess this subset. The number of choices is at most . For each of these guesses (or subsets), we can guess the colors on them. Since we have two colors, and the number of edges outside is at most , there are at most different 2-colorings of these edges. Thus, in total, we will generate
guesses (subsets together with 2-edge-colorings). Now, consider any of these guesses. If it contains at least three edges adjacent to the same vertex, or two edges incident to the same vertex, then we do not consider it. If it contains two edges and incident to the same vertex such that edges have different color, we remove and forbid the corresponding color on the other end-point of and in . If an edge is not adjacent to any other edge in the guess, we simply remove it and forbid its color in its end-points on . Having done this, we get polynomialy many instances of the forest problem with constraints on vertices. By Lemma 3, we can find the largest 2-edge-colorable subgraph respecting the constraints in polynomial time. Thus, we can compare the sizes of all these 2-edge-colorable subgraphs and get a maximum 2-edge-colorable subgraph of in polynomial time. Thus, the total running-time of our algorithm in the second case will be polynomial. The proof is complete. ∎
4 Future work
In this section, we discuss some open problems that will be suitable for future research. For a graph , let be the size of the smallest vertex cover of . Since in any graph , Corollary 1 and Lemma 1 imply that the maximum 2-edge-colorable subgraph problem is FPT with respect to . We would like to ask:
Is the maximum -edge-colorable subgraph problem FPT with respect to ?
The classical 2-approximation algorithm for the vertex cover problem and its analysis imply that for any graph , we have . This inequality means that in any graph , we have:
The classical Gallai theorem west:1996 states that in any graph we have:
Here is the size of the largest independent set of . Since the maximum 2-edge-colorable subgraph problem is FPT with respect to , it is FPT with respect to . We would like to ask:
Can we paramterize the maximum -edge-colorable subgraph problem with respect to ?
Note that in cubic graphs the size of the largest clique and the chromatic number are bounded with four. Thus, combined with Holyer’s result, we have that if , the maximum 2-edge-colorable subgraph problem is not FPT with respect to these two parameters.
Some lines of research that may be worth to investigate concern the generalisation to the weighted case, where each colour can have different weights even in combination with specific edges, adding different classes of constraints among colours, and analyzing them with respect to different graph topologies, like it is done in aloisio:2011 ; aloisio:2019 ; aloisio:2015 .
The second author would like to thank Zhora Nikoghosyan for useful discussions on Hamiltonian graphs.
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In this section we present additional results. In kEdgeColoringFPT , it is shown that for each fixed , the -edge-coloring problem is FPT with respect to the number of maximum degree vertices of the input graph. As we have mentioned previously, the maximum -edge-colorable subgraph problem is harder than -edge-coloring. Thus, one can try to parameterize the latter with respect to the number of vertices of maximum degree. As the following theorem states, if , this is impossible.
If , then the maximum 2-edge-colorable subgraph problem cannot be parameterized with respect to the number of maximum-degree vertices.
Consider the class of graphs from the proof of Theorem 2. Observe that if is the complete graph on four vertices then has five vertices of degree four, which have maximum degree in . On the other hand, if , then is the only vertex of maximum degree. Thus, if we assume that the maximum 2-edge-colorable subgraph problem is FPT with respect to the number of vertices of maximum degree, we will have a polynomial time algorithm for constructing it in the class , hence to say whether . As in the proof of Theorem 2, this implies . ∎
Observe that in the above proof, there is no need for us to join to all the vertices of . Since is cubic we can join to five vertices of . This will lead to the graph , where is the only vertex of degree five, which is maximum. All other vertices are of degree four or three. Thus, the problem remains hard even when the number of maximum degree vertices is one and the maximum degree is five.
Holyer’s result holyer:1981 implies that it is -hard to find a maximum 2-edge-colorable subgraph in cubic graphs. Thus, if , we cannot parameterize the maximum 2-edge-colorable subgraph problem with respect to and . Moreover, in the proof of Theorem 2, we have and , hence in these graphs . Thus, one can say that if , we cannot parameterize the maximum 2-edge-colorable subgraph problem with respect to , too. On the positive side, it turns out that
The maximum 2-edge-colorable subgraph problem is FPT with respect to .
Let be any graph. If , then
Now, if we run the trivial algorithm, its running-time will depend solely on , as we have bounded the number of edges in terms of it. On the other hand, if , then
Thus, by Ore’s classical theorem west:1996 , has a Hamiltonian cycle . Now, if is even, then is a 2-edge-colorable subgraph in . Since in any graph , , we have that is a maximum 2-edge-colorable subgraph in . On the other hand, if is odd, then any matching in has at most edges, hence . Now, if we remove any edge from , then the resulting Hamiltonian path will be a 2-edge-colorable subgraph with edges. Hence it will be a maximum 2-edge-colorable subgraph in . The proof is complete. ∎
Let us note that the proof of Ore’s theorem represents a polynomial time algorithm which actually finds the Hamiltonian cycle. Thus, in the second case of the previous proof, the algorithm will run in polynomial time.
Observe that in any graph , we have the following relationship among vertex, edge connectivity and minimum degree:
Since, the maximum 2-edge-colorable subgraph problem is not FPT with respect to (unless ), Lemma 1 implies that the problem is not FPT with respect to and . Moreover, since in any graph
In the following, we will see that 2-edge-colorable subgraph problem is fixed-parameter tractable respect to the carving width thilikos:2013 of the graph. We first recall some basic definitions that we use in Theorem 6. Let be a graph, and let . As usual, let be the set of edges between and . Clearly, it is an edge cut of . Assume that the vertices of are in 1-to-1 correspondence with the leaves of a sub-cubic tree whose internal vertices all have degree three. This correspondence uniquely defines an edge-weight in the following way: if , and and are the two connected components of , then let be the set of leaves of that are in for . We have . Then the weight . The tree is called a carving of , and is called a carving decomposition of . The width of is the maximum weight over all . The carving-width of , denoted by , is the minimum width over all carving decompositions of . We define if .
An example of carving decomposition is depicted in Figure 5, where the red edges among the leaves correspond to the edges of the decomposed graph in Figure 5. The integers on the edges of the tree are the weights of the decomposition, that is the size of the corresponding edge-cuts.
The maximum 2-edge-colorable subgraph problem is FPT with respect to the carving-width , and can be solved in time.
Let be a graph with a carving decomposition of width , where root is the root of . Denote by the set of colours, where , and are called true colours, while is dummy and means ’not coloured’. Let be the pair , where , and . Also let a function, which takes in input a pair , and returns 1 if , i.e., it is a true colour, and returns 0, otherwise. We use instead of to avoid cluttered notation. Note that only the dummy colour can be used on two or more adjacent edges, because it essentially says that an edge is not coloured. We have extended the set of colours in order to make the proof more readable.
Now, we describe a dynamic programming algorithm to find a maximum 2-edge-colorable subgraph of , which exploits the structure and the properties of carving decomposition. To avoid confusion between the vertices of the graph, and the ones in the tree, we will call nodes the vertices of .
For a node of , denote by the subtree with and all its descendants. Let be the set of vertices corresponding to the leaves in , and let be the set . Define as the subgraph of induced by , and let the subgraph of induced by and the edges in . Let be the optimum value for maximum 2-edge-colorable subgraph problem restricted to , where is a set of edge-colour pairs, one for each edge in , i.e. , that satisfies the following constraints.
The edge is coloured with the colour in the edge-colour pair .
If is a leaf corresponding to a vertex , then one of the following conditions holds:
if there are no two edges in with the same true colour assigned;
If is an interior node of with two sons and , then we compute solving the following maximisation problem, where , i.e., the symmetric difference of and .
In fact, by definition of carving decomposition, we have that , , and that the edges in are the ones in minus the ones in , i.e.,