A Note on Graphs of Linear Rank-Width
We prove that a connected graph has linear rank-width if and only if it is a distance-hereditary graph and its split decomposition tree is a path. An immediate consequence is that one can decide in linear time whether a graph has linear rank-width at most , and give an obstruction if not. Other immediate consequences are several characterisations of graphs of linear rank-width . In particular a connected graph has linear rank-width if and only if it is locally equivalent to a caterpillar if and only if it is a vertex-minor of a path [O-joung Kwon and Sang-il Oum, Graphs of small rank-width are pivot-minors of graphs of small tree-width, to appear in Discrete Applied Mathematics] if and only if it does not contain the co- graph, the Net graph and the -cycle graph as vertex-minors [Isolde Adler, Arthur M. Farley and Andrzej Proskurowski, Obstructions for linear rank-width at most , to appear in Discrete Applied Mathematics].
In their investigations for the recognition of graphs of bounded clique-width  Oum and Seymour introduced the notion of rank-width  of a graph. Rank-width appeared to have several nice combinatorial properties, in particular it is related to the vertex-minor inclusion, and have proven the last years its importance in studying the structure of graphs of bounded clique-width [19, 20, 9, 15]. Linear rank-width is related to rank-width in the same way path-width  is related to tree-width . Indeed, linear rank-width is the linearised version of rank-width and studying graphs of bounded linear rank-width is a first step in studying the structure of graphs of bounded rank-width which is not yet well understood. Not much is known about linear rank-width. The computation of the linear rank-width of forests is investigated in  and it is proved in  that graphs of linear rank-width are vertex-minors of graphs of path-width at most . Ganian defined in  the notion of thread graphs and proved that they correspond exactly to graphs of linear rank-width and authors of  used it to exhibit the set of vertex-minor obstructions for linear rank-width . In this paper we investigate in a different way the structure of graphs of linear rank-width .
Distance hereditary graphs  are a well-known and well-studied class of graphs because of their multiple nice algorithmic properties. They admit several characterisations, in particular they correspond exactly to graphs of rank-width at most  and are the graphs that are totally decomposable with respect to split decomposition . Split decomposition is a graph decomposition introduced by Cunningham and Edmonds and has proved its importance in algorithmic and structural graph theory (see for instance [4, 5, 6, 7, 21] to cite a few). We give in this paper the following characterisation of graphs of linear rank-width which implies all the known characterisations of graphs of linear rank-width .
A connected graph has linear rank width if and only if it is a distance hereditary graph and its split decomposition tree is a path.
A first consequence of this theorem is that we can derive in a more direct way than in  the set of induced subgraph (or vertex-minor of pivot-minor) obstructions for linear rank-width . Another consequence is a simple linear time algorithm for recognising graphs of linear rank-width (the only known one prior to this algorithm is the one that uses logical tools  and is not really practical). Our algorithm gives moreover an obstruction if it exists. Notice that a polynomial time algorithm for recognising graphs of linear rank-width is a consequence of the characterisation in terms of thread graphs given in .
The paper is organised as follows. Some definitions and notations are given in Section 2. In Section 3 we introduce the notion of split decomposition and prove our main theorem. We derive several characterisations and give a simple linear time algorithm (with a certificate) for the recognition of graphs of linear rank-width at most .
For two sets and , we let be the set . We often write to denote the set . For sets and , an -matrix is a matrix where the rows are indexed by elements in and columns indexed by elements in . For an -matrix , if and , we let be the sub-matrix of the rows and the columns of which are indexed by and respectively. We let be the matrix rank-function (the field will be clear from the context).
Our graph terminology is standard, see for instance . A graph is a pair where is the set of vertices and , the set of edges, is a set of unordered pairs of . An edge between and in a graph is denoted by (equivalently ). The subgraph of a graph induced by is denoted by . Two graphs and are isomorphic if there exists a bijection such that if and only if . All graphs are finite and loop-free.
A tree is an acyclic connected graph. In order to avoid confusions in some lemmas, we will call nodes the vertices of trees. The nodes of degree are called leaves. A path is a tree the vertices of which have all degree , except two that have degree . A caterpillar is a tree such that the removal of leaves results in a path.
A graph is distance hereditary if its induced subpaths are isometric . Examples of distance hereditary graphs are trees, cliques, etc. There exist several characterisations of distance hereditary graphs. See for instance [17, Theorem 1] for a summary of some known characterisations of distance hereditary graphs.
The adjacency matrix of a graph is the -matrix over where if and only if . For a graph , let be a linear ordering of its vertices. For each index , we let and . The cutrank of the ordering is defined as
The linear rank-width of a graph is defined as
The linear ordering of the graph in Figure 2 has cutrank . It is worth noticing that if a graph is not connected and its connected components are , then since it suffices to concatenate in any order the linear ordering of optimal cutrank of its connected components.
For a graph and a vertex of , the local complementation at consists in replacing the subgraph induced on the neighbours of by its complement. The resulting graph is denoted by . A graph is locally equivalent to a graph if is obtained from by applying a sequence of local complementations, and is called a vertex-minor of if is isomorphic to an induced subgraph of a graph locally equivalent to . The following relates vertex-minor and linear rank-width.
Proposition 1 ().
Let and be two graphs. If is locally equivalent to , then . If is a vertex-minor of , then .
3. Linear rank-width
We prove in this section our main theorem. Let us make precise some terminologies and notations.
3.1. Split decomposition
Two bipartitions and of a set overlap if for all . A split in a connected graph is a bipartition of the vertex set such that and . A split is strong if there is no other split such that and overlap. Figure 1 shows a schematic view of splits. Notice that not all graphs have a split and those without a split are called prime. We follow  for the definition of a split decomposition tree.
If is a split, then we let and be the graphs with vertex set and respectively where the vertices and are new and called neighbour markers of and respectively, and with edge set
A decomposition of a connected graph is defined inductively as follows: is the only decomposition of size . If is a decomposition of size of , then if has a split , then is a decomposition of size . Notice that the decomposition process must terminate because the new graphs and are smaller than . The graphs of a decomposition are called blocks. If two blocks have neighbour markers, we call them neighbour blocks.
For every decomposition of a connected graph we associate the graph with vertex set and edge set
Edges in between neighbour markers of are called marked edges and the others are called solid edges. One notices that subgraphs of induced by solid edges are blocks of . Observe that each mark edge is an isthmus, and the marked edges form a matching.
Two decompositions and of a connected graph are isomorphic if there exists a graph isomorphism between and which preserves the marked edges, and such that for all . It is worth noticing that a graph can have several non isomorphic decompositions. However, a canonical decomposition can be defined. A decomposition is canonical if and only if: (i) each block is either prime (called prime block), or is isomorphic to a clique of size at least (called clique block) or to a star of size at least (called star block), (ii) no two clique blocks are neighbour, and (iii) if two star blocks are neighbour, then either their markers are both centres or both not centres. The following theorem is due to Cunningham and Edmonds , and Dahlhaus .
Every connected graph has a unique canonical decomposition, up to isomorphism. It can be obtained by iterated splitting relative to strong splits. This canonical decomposition can be computed in time for every graph with vertices and edges.
The canonical decomposition of a connected graph constructed in Theorem 2 is called split decomposition and we will denote it by . Since marked edges of are isthmus and form a matching, if we contract the solid edges in , we obtain a tree called split decomposition tree of and denoted by . For every node of , we denote by the block of the edges of which are contracted to get , and we let be . For an edge of , we denote by the subgraph of induced by where is the subtree of induced by those nodes of such that a path from to does not contain . Notice that for every edge of , is a strong split of . We finish these preliminaries with the following characterisation of distance hereditary graphs.
Theorem 3 ().
A connected graph is distance hereditary if and only if for every node of and for every , the bipartition with such that is a split in , provided that .
3.2. Charaterizing graphs of linear rank-width
It is folklore to verify that the rank-width of a graph is smaller (or equal) than its linear rank-width. Hence, a connected graph of linear rank-width has necessarily rank-width . It is proved in  that a connected graph has rank-width if and only if it is a distance hereditary graph. Therefore, Theorem 1 follows from Propositions 2 and 3 below.
Let be a connected distance hereditary graph such that is a path. Then .
We will show how to turn into a linear ordering of of cutrank . Let us enumerate the nodes of as from left to right. For every , let be any linear ordering of , and let be the concatenation of the orderings . Since is a partition of , is clearly a linear ordering of . We claim that its cutrank is . Indeed let be an index of this ordering and let . Assume without loss of generality that , otherwise we have trivially . Since is a path, then is equal to with for some . By Theorem 3, is a split in , and hence . ∎
The next proposition gives the converse direction of Proposition 2.
Let be a connected graph of linear rank-width . Then is distance hereditary and is a path.
Let be a graph with . Hence, is distance hereditary (see the paragraph before Proposition 2). Let be a linear ordering of of cutrank . It suffices to prove that every strong split of is of the form for some index . Suppose this is not the case and let be a strong split of with for every . Without loss of generality we can assume that and let be the smallest index such that . First of all notice that otherwise and then would not be a split. Therefore, is a split because is a linear ordering of of cutrank . We have that
otherwise would equal ,
because otherwise .
Therefore, and overlap, which contradicts the fact that is a strong split. ∎
Figure 3 shows a graph of linear rank-width exactly .
3.3. Structure and obstructions
We will now discuss about some consequences of Theorem 1, particularly the structure of graphs of linear rank-width .
Let be a graph and let be an edge of . The pivoting of on is the graph [5, 18]. A graph is a pivot-minor of a graph if can be obtained from by a sequence of pivotings and deletions of vertices. It is clear that a pivot-minor is also a vertex-minor. A graph is a vertex-minor (or pivot-minor or induced subgraph) obstruction for linear rank-width if has linear rank-width and every proper vertex-minor (or pivot-minor or induced subgraph) of has linear rank-width . In  the authors gave the induced subgraph (or vertex-minor or pivot-minor) obstructions for linear rank-width . We will explain how to obtain all these obstructions in a more direct way from Theorem 1. The proof of the following is straightforward from Theorem 1.
A graph is a distance hereditary graph obstruction for linear rank-width if and only if its split decomposition tree is a star with three leaves and every connected component of every proper induced subgraph of has a path as split decomposition tree.
From Theorem 1, we can also deduce that if a connected graph has linear rank-width , then for each internal node of , (recall that is the set of vertices of in the block ). Bouchet  characterises exactly distance hereditary graphs such that each internal node of the split decomposition tree contains at least one vertex.
Theorem 4 ().
A distance hereditary graph is locally equivalent to a tree if and only if, for every internal node of , . This tree is moreover unique.
In the following we deduce some interesting characterisations of graphs of linear rank-width we can deduce from Theorem 1 and some results in the literature.
Let be a connected graph. The following statements are equivalent.
is distance hereditary and is a path.
has linear rank-width .
is locally equivalent to a caterpillar.
is a vertex-minor of a path.
does not contain neither the co- graph nor the Net graph nor the -cycle as a vertex-minor.
The equivalence between (1) and (2) is from Theorem 1. The equivalence between (2) and (4) is proved in , but can be easily proved from the equivalence between (2) and (3). Indeed, connected vertex-minors of paths have linear rank-width and since every caterpillar is a vertex-minor of a path, we are done.
The equivalence between (2) and (5) is proved in . Let us explain how to prove it in a more direct way from Proposition 4. It is sufficient to construct the set of induced subgraph obstructions since the set of vertex-minor and pivot-minor obstructions can be derived from that set. The set of induced subgraph obstructions to being a distance hereditary graph is known for a while  and constitutes a subset of the induced subgraph obstruction for linear rank-width . We now explain how to construct the set of distance hereditary induced subgraph obstructions for linear rank-width . Let be a distance hereditary obstruction for linear rank-width . From Proposition 4 its split decomposition tree is a star with three leaves, say centred at with and as leaves. The following three cases describe exactly the canonical split decomposition of .
Case 1: is a clique. Then has exactly three vertices and none of the s is a clique. Moreover, for each the graph has three vertices and is a star.
Case 2: is a star centred at a marker vertex. Then has three vertices. Let us assume without loss of generality that the centre of is the neighbour marker of the marker vertex in . Then is either a clique or a star centred at a marker vertex, and for each the graph has three vertices and is either a clique or a star centred at a vertex of .
Case 3: is a star centred at a vertex of . Then has four vertices and its centre is a vertex of . Moreover, for each the graph has three vertices and is either a clique or a star centred at a vertex of .
From the description of the split decomposition of an induced subgraph obstruction for linear rank-width , one can clearly construct all the induced subgraph obstructions for linear rank-width . In Figure 4 we have recalled the vertex-minor obstructions for linear rank-width . See  for the complete list of pivot-minor and induced subgraph obstructions for linear rank-width .
It remains now to prove the equivalence between (2) and (3). Caterpillars have clearly linear rank-width and are the only trees with paths as split decomposition trees. Now if a graph has linear rank-width , then from Theorem 1 it is a distance hereditary graph and its split decomposition tree is a path and each node is such that . By Theorem 4 it is then locally equivalent to a tree, say . By [5, Theorem 4.4] is also the split decomposition tree of , which concludes the proof. ∎
3.4. Recognition algorithm
Thanks to the previous characterisations, graphs of linear rank-width can be recognised in time .
One can decide in time if a connected graph with vertices and edges have linear rank-width , and if not construct an obstruction. A linear ordering of cutrank can be constructed also in time if it exists.
Let be a graph with vertices and edges. Thanks to  one can check in time if is a distance hereditary graph and if not exhibit an induced subgraph obstruction. From the induced subgraph obstruction one can exhibit, if he prefers, vertex-minor or pivot-minor obstructions.
We will now assume that is distance hereditary. We first construct a split decomposition tree of which can be done in time . By Theorem 1 has linear rank-width if and only if is a path. Since testing whether is a path can be done in time , we can test in time if has linear rank-width at most . Knowing that is a path, one can construct a linear ordering of of cutrank in time by Proposition 2.
We now explain how to exhibit an induced subgraph obstruction if is not a path since from an induced subgraph obstruction one can exhibit a vertex-minor or a pivot-minor obstruction. If is not a path, then there exists an internal node of degree at least three, that can be found in time , and let us choose three of its neighbour nodes say . We need to look at the type of the node .
Case 1: is a clique. Then none of the s is a clique. In each of the graphs take either two non adjacent vertices that are adjacent to vertices in or two adjacent vertices such that exactly one is adjacent to vertices in .
Case 2: is a star centred at a marker vertex. We may assume without loss of generality in this case that there exists a marker vertex in which is a neighbour marker of the centre of . Choose in either two adjacent vertices or two non adjacent vertices, and in each of the graphs , for choose two adjacent vertices such that at least one is adjacent to a vertex in .
Case 3: is a star centred at a vertex of . Take the centre of , and choose in each of the graphs two adjacent vertices such that at least one is adjacent to a vertex in .
One checks easily that in each of the cases above the split decomposition tree of the chosen induced subgraph is a star with three leaves and is minimal with respect to that property, hence is an induced subgraph obstruction. ∎
August 22, 2019
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