Parameterized algorithms are among the most natural approaches to tackle NP-hard optimization problems . In particular, they have been very successful in dealing with so-called edge modification problems on graphs: given as input an arbitrary graph and a parameter , the goal is to transform into a graph with some specific properties (i.e., belonging to a specific graph class ) by adding and/or deleting at most edges. Parameterized algorithms (also called FPT for fixed parameter tractable) aim at a time complexity of type , where is some computable function, hence the combinatorial explosion is restricted to parameter .
When the target class is characterized by a finite family of forbidden induced subgraphs, modification problems are FPT by a result of Cai . Indeed, as long as the graph contains one of the forbidden subgraphs, one can try each possibility to correct this obstruction and branch by recursive calls. On each branch, the budget is strictly diminished, therefore the whole algorithm has a number of calls bounded by some function . The situation is more complicated when the target class is characterized by an infinite family of forbidden induced subgraphs. Nonetheless, a large literature is devoted to edge modification problems towards chordal graphs (where we forbid all induced cycles with at least four vertices) as well as sub-classes of chordal graphs, typically obtained by requiring some fixed set of obstructions, besides the long cycles. Observe that, in this case, the situation remains relatively simple if we restrict ourselves to edge completion problems, where we are only allowed to add edges to the input graphs. Indeed, in this case, if a graph has an induced cycle of length longer than , it cannot be made chordal by adding at most edges. Therefore we can use again the approach of Cai to deal with cycles of length at most and other obstructions, and either the algorithm finds a solution in recursive calls, or we can conclude that we have a no-instance. The cases of edge deletion problems (where we are only allowed to remove edges) and edge editing problems (where we are allowed to both remove edges or add missing edges) are more complicated, since even long cycles can be eliminated by a single edge removal. Therefore more efforts and more sophisticated techniques were necessary in these situations, but several such problems turned out to be FPT [11, 15, 16].
The interested reader can refer to  for a broad and comprehensive survey on parameterized algorithms for edge modification problems.
We focus here on a sub-family of parameterized algorithms, namely on kernelization. The goal of kernelization is to provide a polynomial algorithm transforming any instance of the problem into an equivalent instance where is upper bounded by some function of (in our case we will simply have ), and the size of the new instance is upper bounded by some function . Hence the size of the reduced instance does not depend on the size of the original instance. Kernels are obtained throught a set of reduction rules. While kernelization is possible for all FPT problems (the two notions are actually equivalent), the interesting question is whether a given FPT problem admits polynomial kernels, where the size of the reduced instance is bounded by some polynomial in . Note that, under some complexity assumptions, not all FPT problems admit polynomial kernels [5, 6, 7, 10, 21, 27].
In this paper we focus on Block Graph Editing, Block Graph Deletion and all three variants of modification problems towards striclty chordal graphs. Block graphs are a well-studied subclass of chordal graphs, that are diamond-free (Figure 1). They are moreover the graphs in which every biconnected component induces a clique. Notice that this observation allows for a simple polynomial-time algorithm for Block Graph Completion since one needs to turn every biconnected component of the input graph into a clique. Strictly chordal graphs are another subclass of chordal graphs, also known as block duplicate graphs [25, 26, 20]. They can be obtained from block graphs by repeatedly choosing some cut-vertex and adding a true twin of , that is a vertex adjacent to and all neighbors of . They can also be characterized as dart, gem-free chordal graphs (see Figure 1 and next section) and are thus ptolemaic (i.e. chordal distance-hereditary ). Strictly chordal graphs are also known to be a subclass of 4-leaf power graphs , and a super-class of 3-leaf power graphs .
Kernelization for chordal completion goes back to the ’90s and the seminal paper of Kaplan, Shamir and Tarjan . In this work, the authors provide an vertex-kernel for Chordal Completion (also known as Minimum Fill-In) that was later improved to by a tighter analysis . Since then, several authors addressed completion, deletion and/or editing problems towards sub-classes of chordal graphs, as 3-leaf power graphs , split and threshold graphs , proper interval graphs , trivially perfect graphs [2, 17, 18, 22] or ptolemaic graphs . All these classes have in common that they can be defined as chordal graphs, plus a constant number of obstructions. Several questions remain open, for example it is not known whether chordal deletion or chordal editing admit polynomial kernels .
We first prove that most variants of the considered problems are NP-Complete, the only exception being the Block Graph Completion problem that admits a polynomial-time algorithm. Secondly, we give kernelization algorithms for all these problems, with different bounds. First of all, we begin by illustrating the techniques we will use on the Block Graph Editing problem and obtain an vertex-kernel. This result can be transfered to Block Graph Deletion as well. Next, we present a kernelization algorithm for the Strictly Chordal Editing problem, producing a reduced instance with vertices. Eventually, we discuss how this approach leads to vertex-kernels for both Strictly Chordal Completion and Strictly Chordal Deletion. Above all, our purpose is to exhibit general techniques that might, we hope, be extended to kernelizations for edge modification problems towards other graph classes. Several such algorithms, e.g., [3, 18] share the following feature. Very informally, the target class admits a tree-like decomposition, in the sense that the vertices of any graph can be partitioned into clique modules (having the same neighborhood in ), and these modules can be mapped onto the nodes of a decomposition tree, the structure of the tree describing the adjacencies between modules. Therefore, if an arbitrary graph can be transformed into graph by at most edge additions or deletions, at most modules can be affected by the modifications. By removing the affected nodes from the decomposition tree, we are left with several components (chunks) that correspond, in the initial graph as well as in , to induced subgraphs that may be large but that already belong to the target class. Moreover, these chunks are attached to the rest of graph in a very regular way, through one or two nodes of the decomposition tree. The kernelization algorithms need to analyze these chunks and provide reduction rules, typically by ensuring a small number of nodes in the decomposition tree, plus the fact that each node corresponds to a module of small size.
Both the classes of block graphs and of strictly chordal graphs do not have exactly a tree-like decomposition, but still can be decomposed into structures than can be seen as a generalization of a tree. Our algorithms exploit these informal observations and provide the necessary reduction rules together with the combinatorial analysis for the kernel size.
The paper is organized as follows. We begin with some preliminary results and definitions, including the NP-completeness proofs of aforementioned problems (Section 2). We then describe a kernel with vertices for Block Graph Editing and Block Graph Deletion (Section 3), introducing the techniques that will be adapted to obtain an vertex-kernel for Strictly Chordal Editing (Section 4). We then explain how these techniques produce vertex-kernels for both Strictly Chordal Completion and Strictly Chordal Deletion (Section 5) and then conclude with some perspective and open problems (Section 6).
We consider simple, undirected graphs where denotes the vertex set and the edge set of . We will sometimes use and to clarify the context. Given a vertex , the open neighborhood of is the set . The closed neighborhood of is defined as . Two vertices and are true twins if . Given a subset of vertices , is the set and is the set . We will omit the mention to whenever the context is clear. A subset of vertices is a module if for every vertices , . The subgraph induced by is defined as where . For the sake of readability, given a subset we define as . A subgraph is a connected component of if it is a maximal connected subgraph of . A graph is biconnected if it is still connected after removing any vertex. A subgraph is a biconnected component of if it is a maximal biconnected subgraph of . A set is a separator of if is not connected. Given two vertices and of , the separator is a -separator if and lie in distinct connected components of . Moreover, is a minimal -separator if no proper subset of is a -separator. Finally, a separator is minimal if there exists a pair such that is a minimal -separator.
Given a graph , a critical clique of is a set such that is a clique, is a module and is inclusion-wise maximal under this property.
Notice that is a maximal set of true twins and that the set of critical cliques of any graph partitions its vertex set . This leads to the following definition.
Definition 2 (Critical clique graph).
Let be a graph. The critical clique graph of is the graph with .
Block and strictly chordal graphs
Block graphs are graphs in which every biconnected component is a clique. They can also be characterized as chordal graphs that do not contain diamonds as induced subgraph  (see Figure 1). A natural generalization of block graphs are strictly chordal graphs, also known as block duplicate graphs, that are obtained from block graphs by adding true twins [25, 26, 20].
Theorem 1 ().
Let be a graph. The following properties are equivalent:
is a strictly chordal graph,
The critical clique graph is a block graph,
does not contain any dart, gem or hole as an induced subgraph (see Figure 1),
is chordal and the minimal separators of are pairwise disjoint.
We consider the following problems:
Block Graph Editing
Input: A graph , an integer Question: Does there exist a set of pairs of size at most such that the graph is a block graph, with ?
The Strictly Chordal Editing problem is defined similarly, requiring the graph to be strictly chordal.
The Block Graph Completion and Strictly Chordal Completion (resp. Block Graph Deletion and Strictly Chordal Deletion) problems are defined similarly by requiring to be
disjoint from (resp. included in) edge set . Given a graph
, a set such that the graph belongs to the target graph class is called an edition
of . When is disjoint from (resp. included in ) it is
called a completion (resp. a deletion) of .
For the sake of simplicity we use , and to denote
the resulting graphs in all versions of the problems.
In all cases, is optimal whenever it is minimum-sized. Given an
instance of any of the aforementioned problems, we
say that is a -edition (resp. -completion, -deletion) whenever is an edition (resp. completion, deletion)
of size at most . A vertex is affected by
whenever it is contained in some pair of . We say that an
instance is a yes-instance whenever it admits a -edition (resp. -completion, -deletion). When applied to an instance of the problem, a reduction rule is said to be safe if is a yes-instance if and only if the reduced instance is a yes-instance.
We will use the following result that guarantees that any
clique module of a given graph will remain a clique module in any optimal edition towards some hereditary class of graphs closed under true twin addition, in particular towards strictly chordal graphs.
Lemma 1 ().
Let be an hereditary class of graphs closed under true twin addition. For every graph , there exists an optimal edition (resp. completion, deletion) into a graph of such that for any two critical cliques and ’ either or .
Notice in particular that Lemma 1 implies that whenever the target graph class is hereditary and closed under true twin addition one may reduce the size of critical cliques to . We shall see Section 3 that this result does not apply to block graphs. However, we will still be able to bound the size of critical cliques by .
We now turn our attention to the notion of join composition (operation ) between two graphs. This operation will play an important part in the proofs of our reduction rules for all considered problems. Let and be two vertex-disjoint graphs and let . The join composition of and on and , denoted , is the graph .
Let and be two disjoint chordal graphs, and be cliques of respectively and and . Then graph is chordal, and any minimal separator of is also a minimal separator of or , or for some and is not a maximal clique of .
is chordal since any cycle of is either contained in one of , or, if it intersects both of them, contains at least three vertices of , which is a clique in . In both cases, if the cycle has four or more vertices then it contains a chord.
Any minimal separator of chordal graph is the intersection of two maximal cliques of , and moreover is a minimal -separator for any , — in particular, is strictly included in each of the cliques (see, e.g., Theorem 7 and Lemma 5 in ). By construction of , any of its maximal cliques is either a maximal clique of or of , or it is exactly .
If , then graph is not connected. This only happens if one of the graphs is not connected, or if one of the sets is empty. In the first case, is a minimal separator of , in the second case we have that and is not a maximal clique of , so the conclusion holds. From now on, we assume that is not empty.
Consider a first case when none of equals . Then both are contained in the same , for (otherwise their intersection would be empty). Assume w.l.o.g they are both in , we claim that is a minimal separator of . Take and . Since separates and in , it also separates them in , which is a subgraph of . But is in the common neighborhood of and in , thus is minimal among the -separators of , and the conclusion follows.
We are left with the case when and is contained in one of the two graphs , say in . Therefore . If this inclusion is strict, we claim that is also a minimal separator of . Indeed is a minimal separator in for any and , therefore we can choose . Again separates and in the subgraph of , and since is in the common neighborhood of and in , it is necessarily a minimal separator of . We deduce that, if is not a minimal separator of , we must have . We conclude that is strictly contained in the clique of , which proves our lemma. ∎
Let and be two disjoint block graphs and let . The graph is a block graph if for , is a single vertex or a maximal clique of .
Observe that minimal separators of block graphs are single vertices since the biconnected components of block graphs are cliques. From Lemma 2, is chordal and its minimal separators are the ones of and plus eventually or if they are not maximal cliques. Hence minimal separators of are single vertices and it follows that is a block graph. ∎
Let and be two disjoint strictly chordal graphs and let . The graph is strictly chordal if for , is a critical clique, a maximal clique or intersects exactly one maximal clique of .
From Lemma 2, is chordal and its minimal separators of are the ones of and plus eventually or if they are not maximal cliques. If intersects exactly one maximal clique, , it is clear that it does not intersect any minimal separator of . If is a critical clique, we claim that either is a minimal separator of or does not intersect any minimal separator of . Indeed, if there is a minimal (clique) separator of such that , then since is a critical clique, there exist and such that is adjacent to exactly one of and . Suppose w.l.o.g. that is adjacent to and not to , then is in a minimal -separator that intersects and is not equal to it. This is a contradiction since is strictly chordal and from Theorem 1 its minimal separators are pairwise disjoint. It follows that the minimal separators of are pairwise disjoint and by Theorem 1 is strictly chordal. ∎
In the proofs of some rules for the kernel for Strictly Chordal Editing and its variants, we will have to do a join composition of some graphs on a graph with the same set of vertices . The following observation guarantees us that this operation is possible.
Let be disjoint strictly chordal graphs and for each let be a critical clique, a maximal clique or a set that intersects exactly one maximal clique of . If is a critical clique or intersects exactly one maximal clique of , then is a critical clique or intersects exactly one maximal clique of , implying that the graph is strictly chordal.
Spanning block subgraph
The following result is crucial to bound the size of the kernels for all variants of the studied modification problems.
Let be a connected block graph, a set and be a minimal connected induced subgraph of that spans all vertices of . Denote by the set of vertices of degree at least in . The following holds:
The subgraph is unique,
The graph contains at most connected components.
A convenient way to represent the tree-like structure of block graph is its block-cut tree . Recall that the block-cut tree of a graph has two types of nodes: the block nodes correspond to blocks of (i.e, biconnected components which, in our case, are precisely the maximal cliques of ) and the cut nodes correspond to cut-vertices of . We put an edge between a cut node and a block node in if the corresponding cut-vertex belongs to the corresponding block of .
Each vertex of that is not a cut-vertex belongs to a unique block of . Therefore we can map each vertex of on a unique node of as follows: if is a cut-vertex, we map it on its corresponding cut node in , otherwise we map it on the block node corresponding to the unique block of containing . Observe that, for any two vertices , any (elementary) path from to in corresponds to the path from to in . In particular, contains all vertices corresponding to cut nodes of , therefore must contain all these vertices. Altogether, the vertices of are precisely the vertices of , plus all cut-vertices of corresponding to the cut nodes of such paths, implying the unicity of .
Let denote the subtree of spanning all nodes of . We count the vertices of , so let be such a vertex. By the previous observation, it is a cut-vertex of , so is a cut node of . Let be a neighbor of in . By construction of and , there is a block node adjacent to in , such that and are in the maximal clique of corresponding to node . Moreover, is in , or is a cut-vertex of such that the cut node is adjacent to in . Hence we have:
is of degree at least 3 in , or
is the neighbor of a block node of degree at least three in , or, if none of these hold, then
has exactly two neighbors and in , the corresponding maximal cliques of contain at least vertices of , where plus the degrees of and in is at least three.
Let be the number of leaves of . Observe that for any leaf of , there is some such that . Choose for each leaf a unique vertex such that , we call a leaf vertex. Note that the number of vertices corresponding to the first two items of the above enumeration is upper bounded by . Indeed, is incident to an edge of , having an end node of degree at least 3. One can easily check that, in any tree of leaves, the number of such edges is bounded by (this can be shown by induction on the number of leaves of the tree, adding a new leaf node at a time). We count now the vertices of the third type. By the third item, has at least one neighbor in graph , such that is not a leaf vertex. Observe that can be in the neighborhood of at most two vertices of this third type. Altogether it follows that .
To prove the third item, the number of components of , we visualize again the situation in . Recall that any node of degree at least three in is either a cut node, in which case it is in (the first case of the enumeration above) or a block node, but then all its neighbors in correspond to vertices of (the second case of the enumeration above). Therefore the components of correspond to the components of after removal of all nodes of degree at least . Hence the number of such components is upper bounded by . Removing the leaf vertices of does not increase the number of components, and the removal of each other vertex of increases the number of components by at most one. Thus has at most components, concluding the proof of our lemma. ∎
2.1 Hardness results
We first show the NP-completeness of Block Graph Deletion and Block Graph Editing by giving a reduction from Cluster Deletion and Cluster Editing, known to be NP-complete [28, 34, 30]. Notice that Block Graph Completion admits a polynomial-time algorithm due to the very nature of block graphs: any biconnected component of the input graph (computable in linear time ) must be turned into a clique.
Block Graph Deletion and Block Graph Editing are NP-Complete.
We first prove that Block Graph Deletion is NP-Complete by providing a reduction from Cluster Deletion. A graph is a cluster graph if it does not contain any induced path on three vertices (so-called ). Given an instance of Cluster Deletion, we construct an instance of Block Graph Deletion by adding a universal vertex adjacent to all vertices of . Let be the produced instance.
We show that the graph admits a -deletion into a cluster graph if and only if admits a -deletion into a block graph. Suppose first that there is a -deletion of into a cluster graph. The graph is a graph without any as induced subgraph. Now consider the graph . By construction contains no , so is chordal and contains no diamond, thus is a block graph.
Now suppose that there exists a -deletion of into a block graph. We claim that either does not affect the universal vertex or we can construct a -deletion from that does not affect . To support this claim, assume that contains at least one edge incident to .
First, suppose that is not connected. Let be the connected component that contains . For any other connected component of , take a vertex and construct the deletion set . By Lemma 3, is a block graph, and the other connected components of are unchanged. Hence is a block graph and since , is a -deletion. We iterate this construction until we get an intermediate solution such that is connected.
If does not affect we set and we are done. Otherwise there is at least one vertex not adjacent to in . Since is connected we can choose at distance exactly from . Since is a block graph we have . Consider the biconnected component containing and , and let . Since is a block graph, (and thus ) is a clique. Let and be the connected components of that contain the vertices and respectively, they are block graphs by heredity. By Lemma 3 the graph is a block graph. Observe that since is an universal vertex and let be the deletion set such that . By construction we can observe that , hence is a -deletion of into a block graph. We iterate this construction until we get a -deletion of that does not affect .
Finally, does not contains any or else there would be a diamond in with the universal vertex , thus is a cluster graph.
The same reduction can be done for Block Graph Editing, one needs only to observe that there are no added edges incident to , and we can use the same argument as above on the deleted edges: if there is no deleted edge incident to we are done, otherwise we construct another set of deleted edges of equal size, but containing fewer edges incident to . If in the construction we have to delete an edge that was added by the edition, it is clear that removing this edge from the edition set result in a strictly smaller edition set. ∎
The NP-completeness of Strictly Chordal Completion follows directly from the proof of NP-completeness of 3-Leaf Power Completion from [15, Theorem 3].
Strictly Chordal Completion is NP-complete.
We show the NP-completeness of Strictly Chordal Editing and Strictly Chordal Deletion by giving a reduction from Cluster Editing and Cluster Deletion, respectively.
Strictly Chordal Editing and Strictly Chordal Deletion are NP-complete.
Given an instance of Cluster Editing, we construct an instance of Strictly Chordal Editing by adding a clique of size adjacent to all vertices of , and for each vertex in , vertices adjacent only to . Let be the produced instance. We show that the graph admits a -edition into a cluster graph if and only if admits a -edition into a strictly chordal graph. Suppose first that there is a -edition of into a cluster graph. The graph is a graph without any as induced subgraph. Consider the graph . By construction contains no , so is chordal and contains neither gems nor darts since these obstructions contain an induced and a vertex adjacent to every vertex of this . By Theorem 1, it follows that is strictly chordal. Now suppose that there exists a -edition of into a strictly chordal graph . We claim that is a cluster graph. By contradiction, suppose that contains a where are the ends of the path. Then, there exist such that forms a dart in , contradicting that is strictly chordal, and thus is a cluster graph.
The same reduction can be done from Cluster Deletion to Strictly Chordal Deletion. This concludes the proof. ∎
3 Kernelization algorithm for Block Graph Editing
We begin this section by providing a high-level description of our kernelization algorithm. A similar technique will be applied for modification problems towards strictly chordal graphs but on the critical clique graph rather than the original one. Let us consider a positive instance of Block Graph Editing, a suitable solution and . Since , we know that at most vertices of may be affected vertices. Let be the set of such vertices, the minimum induced subgraph of that spans all vertices of and the set of vertices of degree at least in . From Lemma 5 we have . We will define the notion of BG-branch, corresponding to subgraphs of that induce block graphs. We will focus our interest on two types of BG-branches: the ones that are connected to the rest of the graph by only one (cut) vertex, called -BG-branches, and the ones that are connected to the rest of the graph by exactly two non adjacent vertices, called -BG-branches. We will prove that we can keep only the cut-vertices of -BG-branches and reduce the -BG-branches containing more than vertices. We will see that there are two kinds of connected components remaining in the graph , the ones adjacent to maximal cliques of vertices of , corresponding to critical cliques (which can be bounded to a linear number of vertices) and the ones adjacent to two non adjacent vertices of (which correspond to -BG-branches in ). Since is a block graph, we know that there is at most maximal cliques of vertices of . From Lemma 5 we have that there are at most connected components in , and each one corresponds to a -BG-branch in the graph . It remains that contains a linear number of connected components, each one containing vertices. Altogether, the graph contains vertices.
3.1 Classical reduction rules
We first give classical reduction rules when dealing with modification problems. Notice that Lemma 1 cannot be directly applied to Block Graph Editing and Block Graph Deletion since block graphs are not closed under true twin addition. Hence one cannot directly reduce the size of critical cliques to as mentioned Section 2. To circumvent this issue, we provide a slightly weaker reduction rule.
Let be a block connected component of . Remove from .
Let be a set of true twins of such that . Remove arbitrary vertices from in .
Rule 3.1 is safe since block graphs are hereditary and closed under disjoint union.
We now turn our attention to Rule 3.2. Let denote the graph obtained from the removal of arbitrary vertices from in and . Assume first that admits a -edition . Observe that is a -edition of by heredity. Conversely, assume that admits a -edition and let . Notice that since affects at most vertices and since , there exist two unaffected vertices and in . We claim that is a block graph. Assume for a contradiction that contains an obstruction . By construction, must intersect since otherwise would exist in . Moreover, notice that , otherwise would contain a clique module with or vertices, which is impossible in a cycle or a diamond. Hence assume first that : in this case, since is unaffected we have and thus the set induces an obstruction in , a contradiction. Similarly, if the set induces an obstruction in , leading once again to a contradiction. Finally, since the biconnected components can be computed in time , since block graphs can be recognized in time and since true twins can be detected in time (using modular decomposition ), both rules can be computed in linear time. ∎
3.2 Reducing Block Graph branches
Let be a graph, a BG-branch is a connected induced subgraph of such that is a block graph. A vertex is an attachment point of if . A BG-branch is a -BG-branch if it has exactly attachment points. We denote the subgraph of in which all attachment points have been removed. A -BG-branch is clean if is connected and if the two attachment points are not adjacent. For a clean -BG-branch , a min-cut of is a set of edges of minimum size such that and are not in the same connected component of and is the size of a min-cut of .
Let be a -BG-branch of such that is connected and not empty. Remove from .
Rule 3.3 is safe.
Let be such a -BG-branch and . Let be a -edition of into a block graph. Since the class of block graphs is hereditary, is a block graph. It follows that is a block graph and is a -edition of into a block graph.
Conversely, let be a -edition of into a block graph. Let be the attachment point of in . Observe that since is connected and not empty, is not a cut-vertex of the block graph and its neighborhood in is either a maximal clique or a single vertex. Since and are block graphs and is a single vertex or a maximal clique of , Lemma 3 ensures that is a block graph. We can observe that , hence is a -edition of into a block graph. This concludes the proof. ∎
Let be a yes-instance of Block Graph Editing reduced by Rule 3.3, and a clean -BG-branch of . If then there exists a -edition of into a block graph such that the set of pairs of containing a vertex of is either empty or exactly a min-cut of .
Let be a clean -BG-branch in with attachment points such that . We can observe that and are maximal cliques or single vertices since is connected. Since is reduced by Rule 3.3, we can observe that the cut-vertices of are contained in any path from to . Indeed, if there is a cut-vertex that is not in , then and are in the same connected component of , implying that there is a connected component of containing neither nor . Since is a -BG-branch, , hence induces a -BG-branch of with attachment point , a contradiction. Notice in particular that there is a unique shortest path between and that contain all cut-vertices of . Moreover, since the only cut-vertices of are the ones in , a vertex of that is not in lies in exactly one maximal clique of containing two consecutive vertices of .
Let be a -edition of into a block graph, and . First consider the case in which and are in two different connected components in . Let be the two connected components of containing and , respectively, and the remaining of . It follows from Lemma 3 that the graph resulting from the disjoint union of and is a block graph. Let be the edition such that . By construction, is a subset of with no pair containing a vertex of .
Assume now that and are in the same connected component in and let denote a shortest path between them. We first consider the case where there is a path (not necessarily induced) from to in that still exists in . Since is a clean -BG-branch of reduced by Rule 3.3, we know by the previous arguments that every cut-vertex of is in . Note that all vertices in belong to the same biconnected component of , which is a clique. Hence we can consider w.l.o.g. that is the path containing exactly and the cut-vertices of , which is the shortest path in between and . Since the vertices of induce a path in (on at least vertices since and are not adjacent) and a clique in , then for every , we have . Any vertex is adjacent in to two vertices and if remains in the same biconnected component as and in , then or . Otherwise must contain or . In any case, contains at least one pair for each vertex (see Figure 2). It follows from the previous arguments that . However, since we have , contradicting that is a solution, therefore there is no such path .
We can assume that if and are in the same component of , is not connected and contains an edge-cut of . Let be a min-cut of and consider (resp. ) the connected component containing (resp. ) in . Both and are induced subgraphs (one of them possibly empty) of and block graphs by heredity. By Lemma 3, is a block graph (see Figure 3).
Let the edition such that , we have , since is a min-cut of , and is a -edition of .
In all cases, there exists a -edition of into a block graph such that the set pairs of containing a vertex of is either empty or exactly a min-cut of . This concludes the proof. ∎
Let be an instance of Strictly Chordal Editing and be a clean -BG-branch of containing at least vertices with attachment points . Remove from and add a vertex adjacent to and , a clique of size adjacent to and and a clique of size adjacent to and .
Rule 4 is safe.
Let be the graph obtained from an application of Rule 3.4 on the -BG-branch of graph with attachment points and the vertices introduced by this rule. Observe that is a clean -BG-branch of , and .
Let be a -edition of that satisfies Lemma 11 and . If contains a min-cut of , observe that and consider the connected component of containing , where is a min-cut of . By Lemma 3 the graph is a block graph. Let be the edition such that . Since , by construction , therefore is a -edition of . If does not contain a min-cut of , then no vertices of is affected and are not in the same connected component of . Let and be the connected components of that contain respectively and and the remaining components of . By Lemma 3 the graph corresponding to the disjoint union of