A Note on Graphs of Linear RankWidth
Abstract.
We prove that a connected graph has linear rankwidth if and only if it is a distancehereditary 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 rankwidth at most , and give an obstruction if not. Other immediate consequences are several characterisations of graphs of linear rankwidth . In particular a connected graph has linear rankwidth if and only if it is locally equivalent to a caterpillar if and only if it is a vertexminor of a path [Ojoung Kwon and Sangil Oum, Graphs of small rankwidth are pivotminors of graphs of small treewidth, 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 vertexminors [Isolde Adler, Arthur M. Farley and Andrzej Proskurowski, Obstructions for linear rankwidth at most , to appear in Discrete Applied Mathematics].
1. Introduction
In their investigations for the recognition of graphs of bounded cliquewidth [8] Oum and Seymour introduced the notion of rankwidth [20] of a graph. Rankwidth appeared to have several nice combinatorial properties, in particular it is related to the vertexminor inclusion, and have proven the last years its importance in studying the structure of graphs of bounded cliquewidth [19, 20, 9, 15]. Linear rankwidth is related to rankwidth in the same way pathwidth [22] is related to treewidth [23]. Indeed, linear rankwidth is the linearised version of rankwidth and studying graphs of bounded linear rankwidth is a first step in studying the structure of graphs of bounded rankwidth which is not yet well understood. Not much is known about linear rankwidth. The computation of the linear rankwidth of forests is investigated in [2] and it is proved in [16] that graphs of linear rankwidth are vertexminors of graphs of pathwidth at most . Ganian defined in [14] the notion of thread graphs and proved that they correspond exactly to graphs of linear rankwidth and authors of [1] used it to exhibit the set of vertexminor obstructions for linear rankwidth . In this paper we investigate in a different way the structure of graphs of linear rankwidth .
Distance hereditary graphs [3] are a wellknown and wellstudied class of graphs because of their multiple nice algorithmic properties. They admit several characterisations, in particular they correspond exactly to graphs of rankwidth at most [19] and are the graphs that are totally decomposable with respect to split decomposition [10]. 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 rankwidth which implies all the known characterisations of graphs of linear rankwidth .
Theorem 1.
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 [1] the set of induced subgraph (or vertexminor of pivotminor) obstructions for linear rankwidth . Another consequence is a simple linear time algorithm for recognising graphs of linear rankwidth (the only known one prior to this algorithm is the one that uses logical tools [9] 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 rankwidth is a consequence of the characterisation in terms of thread graphs given in [14].
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 rankwidth at most .
2. Preliminaries
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 submatrix of the rows and the columns of which are indexed by and respectively. We let be the matrix rankfunction (the field will be clear from the context).
Our graph terminology is standard, see for instance [13]. 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 loopfree.
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 [3]. 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 rankwidth 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 vertexminor of if is isomorphic to an induced subgraph of a graph locally equivalent to . The following relates vertexminor and linear rankwidth.
Proposition 1 ([19]).
Let and be two graphs. If is locally equivalent to , then . If is a vertexminor of , then .
3. Linear rankwidth
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 [7] 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 [10], and Dahlhaus [11].
Theorem 2 ([10, 11]).
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 ([10]).
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 rankwidth
It is folklore to verify that the rankwidth of a graph is smaller (or equal) than its linear rankwidth. Hence, a connected graph of linear rankwidth has necessarily rankwidth . It is proved in [19] that a connected graph has rankwidth if and only if it is a distance hereditary graph. Therefore, Theorem 1 follows from Propositions 2 and 3 below.
Proposition 2.
Let be a connected distance hereditary graph such that is a path. Then .
Proof.
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.
Proposition 3.
Let be a connected graph of linear rankwidth . Then is distance hereditary and is a path.
Proof.
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

and ,

otherwise would equal ,

because otherwise .
Therefore, and overlap, which contradicts the fact that is a strong split. ∎
Figure 3 shows a graph of linear rankwidth exactly .
3.3. Structure and obstructions
We will now discuss about some consequences of Theorem 1, particularly the structure of graphs of linear rankwidth .
Let be a graph and let be an edge of . The pivoting of on is the graph [5, 18]. A graph is a pivotminor of a graph if can be obtained from by a sequence of pivotings and deletions of vertices. It is clear that a pivotminor is also a vertexminor. A graph is a vertexminor (or pivotminor or induced subgraph) obstruction for linear rankwidth if has linear rankwidth and every proper vertexminor (or pivotminor or induced subgraph) of has linear rankwidth . In [1] the authors gave the induced subgraph (or vertexminor or pivotminor) obstructions for linear rankwidth . 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.
Proposition 4.
A graph is a distance hereditary graph obstruction for linear rankwidth 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 rankwidth , then for each internal node of , (recall that is the set of vertices of in the block ). Bouchet [5] characterises exactly distance hereditary graphs such that each internal node of the split decomposition tree contains at least one vertex.
Theorem 4 ([5]).
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 rankwidth we can deduce from Theorem 1 and some results in the literature.
Corollary 1.
Let be a connected graph. The following statements are equivalent.

is distance hereditary and is a path.

has linear rankwidth .

is locally equivalent to a caterpillar.

is a vertexminor of a path.

does not contain neither the co graph nor the Net graph nor the cycle as a vertexminor.
Proof.
The equivalence between (1) and (2) is from Theorem 1. The equivalence between (2) and (4) is proved in [16], but can be easily proved from the equivalence between (2) and (3). Indeed, connected vertexminors of paths have linear rankwidth and since every caterpillar is a vertexminor of a path, we are done.
The equivalence between (2) and (5) is proved in [1]. 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 vertexminor and pivotminor obstructions can be derived from that set. The set of induced subgraph obstructions to being a distance hereditary graph is known for a while [3] and constitutes a subset of the induced subgraph obstruction for linear rankwidth . We now explain how to construct the set of distance hereditary induced subgraph obstructions for linear rankwidth . Let be a distance hereditary obstruction for linear rankwidth . 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 rankwidth , one can clearly construct all the induced subgraph obstructions for linear rankwidth . In Figure 4 we have recalled the vertexminor obstructions for linear rankwidth . See [1] for the complete list of pivotminor and induced subgraph obstructions for linear rankwidth .
It remains now to prove the equivalence between (2) and (3). Caterpillars have clearly linear rankwidth and are the only trees with paths as split decomposition trees. Now if a graph has linear rankwidth , 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 rankwidth can be recognised in time .
Theorem 5.
One can decide in time if a connected graph with vertices and edges have linear rankwidth , and if not construct an obstruction. A linear ordering of cutrank can be constructed also in time if it exists.
Proof.
Let be a graph with vertices and edges. Thanks to [12] 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, vertexminor or pivotminor obstructions.
We will now assume that is distance hereditary. We first construct a split decomposition tree of which can be done in time [11]. By Theorem 1 has linear rankwidth 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 rankwidth 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 vertexminor or a pivotminor 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
References
 Isolde Adler, Arthur M. Farley, and Andrzej Proskurowski, Obstructions for linear rankwidth at most , To appear in Discrete Applied Mathematics (2013).
 Isolde Adler and Mamadou Moustapha Kanté, Linear rankwidth and linear cliquewidth of trees, WG, 2013.
 HansJürgen Bandelt and Henry Martyn Mulder, Distancehereditary graphs, Journal of Combinatorial Theory, Series B 41 (1986), no. 2, 182–208.
 André Bouchet, Digraph decompositions and eulerian systems, SIAM Journal on Algebraic and Discrete Methods 8 (1987), no. 3, 323–337.
 by same author, Transforming trees by successive local complementations, Journal of Graph Theory 12 (1988), no. 2, 195–207.
 by same author, Circle graph obstructions, Journal of Combinatorial Theory, Series B 60 (1994), no. 1, 107–144.
 Bruno Courcelle, The monadic secondorder logic of graphs XVI : Canonical graph decompositions, Logical Methods in Computer Science 2 (2006), no. 2.
 Bruno Courcelle and Stephan Olariu, Upper bounds to the clique width of graphs, Discrete Applied Mathematics 101 (2000), no. 13, 77–114.
 Bruno Courcelle and Sangil Oum, Vertexminors, monadic secondorder logic, and a conjecture by seese, Journal of Combinatorial Theory, Series B 97 (2007), no. 1, 91–126.
 William H. Cunnigham and Jack Edmonds, A combinatorial decomposition theory, Canadian Journal of Mathematics 32 (1980), 734–765.
 Elias Dahlhaus, Parallel algorithms for hierarchical clustering, and applications to split decomposition and parity graph recognition, Journal of Algorithms 36 (2000), no. 2, 205–240.
 Guillaume Damiand, Michel Habib, and Christophe Paul, A simple paradigm for graph recognition: application to cographs and distance hereditary graphs., Theoretical Computer Science 263 (2001), no. 12, 99–111.
 Reinhard Diestel, Graph Theory (Graduate Texts in Mathematics), Springer, 2005.
 Robert Ganian, Thread graphs, linear rankwidth and their algorithmic applications, IWOCA (Costas S. Iliopoulos and William F. Smyth, eds.), Lecture Notes in Computer Science, vol. 6460, Springer, 2010, pp. 38–42.
 Petr Hlinený and Sang il Oum, Finding branchdecompositions and rankdecompositions, SIAM J. Comput. 38 (2008), no. 3, 1012–1032.
 O joung Kwon and Sang il Oum, Graphs of small rankwidth are pivotminors of graphs of small treewidth, To appear in Discrete Applied Mathematics (2013).
 Mamadou Moustapha Kanté and Michaël Rao, Directed rankwidth and displit decomposition, WG (Christophe Paul and Michel Habib, eds.), Lecture Notes in Computer Science, vol. 5911, 2009, pp. 214–225.
 SangIl Oum, Graphs of bounded rank width, Ph.D. thesis, Princeton University, 2005.
 by same author, Rankwidth and vertexminors., Journal of Combinatorial Theory, Series B 95 (2005), no. 1, 79–100.
 SangIl Oum and Paul D. Seymour, Approximating cliquewidth and branchwidth., Journal of Combinatorial Theory, Series B 96 (2006), no. 4, 514–528.
 Michaël Rao, Solving some npcomplete problems using split decomposition, Discrete Applied Mathematics 156 (2008), no. 14, 2768–2780.
 Neil Robertson and Paul D. Seymour, Graph minors. I. excluding a forest, Journal of Combinatorial Theory, Series B 35 (1983), no. 1, 39–61.
 by same author, Graph minors. II. algorithmic aspects of treewidth, Journal of Algorithms 7 (1986), no. 3, 309–322.