Betti Numbers of Prodsimplicial Complexes for Directed Graphs with Applications to Word Reductions

A directed graph (digraph for short) $G$ is a 4-tuple $(V,E,\iota,\tau)$ where $V$ is a finite set, $E\subseteq V\times V\setminus\{(x,x)\ |\ x\in V\}$ and $\iota,\tau$ are maps from $E$ to $V$. Elements of $V$ and $E$ are called vertices and edges, respectively, and we use $V(G)$ and $E(G)$ to denote the sets of vertices and edges of a given digraph $G$. The map $\iota$ indicates the source (also called initial) vertex of each edge, and similarly $\tau$ indicates the target (also called terminal) vertex. Directly from the definition, it follows that the graphs considered have no loops or parallel edges. An edge $e$ with source $u$ and target $v$ is denoted ¹¹1To distinguish between ordered pairs in the Cartesian product of two sets and directed edges, we use $[u,v]$ to denote the ordered pair $(u,v)$ when it corresponds to an edge, so that $[(u_{1},v_{1}),(u_{2},v_{2})]$ denotes the directed edge $(u_{1},v_{1})\rightarrow(u_{2},v_{2})$ between vertices $(u_{1},v_{1})$ and $(u_{2},v_{2})$. with $e=[u,v]$. A source of $G$ is a vertex $s$ such that $\tau(e)\neq s$ for every edge $e\in E$. Similarly, a target of $G$ is a vertex $t$ such that $\iota(e)\neq t$ for every edge $e\in E$.

For ease of notation, we refer to a digraph as $G=(V,E)$, omitting the maps $\iota$ and $\tau$ whenever clear from the context. We refer the reader to [11] for other elementary definitions in graph theory not explicitly recalled here.

We use the definition of Cartesian product as in [24].

Note that $G_{1}\square G_{2}\cong G_{2}\square G_{1}$, as $[(u_{1},v_{1}),(u_{2},v_{2})]\in E(G_{1}\square G_{2})$ if and only if $[(v_{1},u_{1}),(v_{2},u_{2})]\in E(G_{2}\square G_{1})$ so that the Cartesian product is commutative modulo isomorphism. Moreover, as proved in [18], it is also associative. As a consequence, when dealing with the Cartesian product of more than two graphs we write $G=((\ldots((G_{1}\square G_{2})\square G_{3})\ldots)\square G_{k})$ without including nested parentheses. Similarly, we write $V(G)=\{((v_{1},\ldots,v_{k-1}),v_{k})\ |\ v_{i}\in V(G_{i})\text{ for }1\leq i\leq k\}$ without nested parentheses.

The Cartesian product $G_{1}\square G_{2}$ creates one copy (referred to as a $G_{1}$-layer in [18]) of $G_{1}$ for every vertex in $G_{2}$, and vice versa. It can be shown that the Cartesian product $G_{1}\square G_{2}$ is consistently directed if and only if both factors are consistently directed [12]. By associativity, this result extends to Cartesian products of several factors.

Vertices and edges in a graph can be considered as 0-dimensional and 1-dimensional cells respectively, therefore a graph can be endowed with a richer structure by attaching higher dimensional cells at instances of particular subgraphs.

In this section we present the main definitions of prodsimplicial complexes introduced in [20] for directed graphs. Other complexes, such as the $p$-path complexes, called $p$-path complexes, have also been considered in [15, 14]. Our motivation comes from a biological process where two paths of length 2 connecting two vertices correspond to two independent pathways of DNA recombination. For such path pairs, we attach solid square faces so that the two pathways are regarded as equivalent.

Note that the source and target of the edges of $\Delta^{n}$ are induced by the total order on the set of vertices $V(\Delta^{n})=\{v_{0},v_{1},\ldots,v_{n}\}$. It follows that $\Delta^{n}$ is consistently directed, with source $v_{0}$ and target $v_{n}$. Moreover, the total order on the vertices guarantees that all subgraphs of $\Delta^{n}$ induced by a subset of $V(\Delta^{n})$ are also simplicial digraphs. Also, any two simplicial digraphs on the same number of vertices are isomorphic as directed graphs. In general, $\Delta^{n}$ can be obtained from $\Delta^{n-1}$ by adding the new vertex $v_{n}$ along with the edges $[v_{i},v_{n}]$ for $i=0,\ldots,n-1$.

In other works, simplicial digraphs are referred to as directed cliques [23, 21] or transitive tournaments [8]. Simplicial digraphs are prime with respect to the Cartesian product. Thanks to the unique prime decomposition in [13], prodsimplicial cells as described below are well defined.

Since the Cartesian product of graphs is commutative up to isomorphism, we assume that in a prodsimplicial cell $P=\operatorname*{{\square}}_{i=1}^{k}\Delta^{n_{i}}_{i}$ the order of the factors is such that $n_{1}\geq n_{2}\geq\cdots\geq n_{k}$. From this point on, a simplex will refer exclusively to a prodsimplicial cell with a single simplicial digraph factor unless otherwise specified. Given the correspondence between simplicial digraph factors and simplices, we abuse the notation and write $P=\operatorname*{{\square}}_{i=1}^{k}\Delta^{n_{i}}_{i}$ instead of $\prod_{i=1}^{k}\Delta^{n_{i}}_{i}$. Also note that the indices are necessary because the dimensions of the factors in the definition of $P$ may not be distinct. However, we omit the subscripts for brevity where it causes no confusion.

We can naturally define the geometric realization of a simplicial digraph as for standard simplices. From this geometric realization, prodsimplicial $N$-cells can be viewed as subsets of $\mathbb{R}^{N}$ that inherit the product topology.

The notion of prodsimplicial cell was first introduced for direct products of undirected graphs by Kozlov in [20]. We choose Cartesian products to obtain a complex that combines some features of both simplicial and cubical complexes while the cells remain consistently directed. Complexes comprised of prodsimplicial cells are also used in knot theory [6, 5].

In this section we describe a boundary operator for prodsimplicial cells, which allows us to compute homology groups for prodsimplicial cell complexes. Although a chain complex structure is defined on a prodsimplicial complex as a special case of CW-complexes, we present an explicit boundary operator on prodsimplicial cells using the product rule for computational purposes.

Given a prodsimplicial cell complex $\Gamma$, a prodsimplicial $N$-chain group on $\Gamma$, denoted $C_{N}(\Gamma)$, is the free abelian group generated by oriented $N$-dimensional prodsimplicial cells of $\Gamma$. Its typical element, a prodsimplicial $N$-chain, is a finite formal linear combination of $N$-dimensional prodsimplicial cells with integer coefficients. In the Cartesian product of a simplex and a chain, the product distributes over the sum; that is

Recall that the boundary operator for simplices is defined by

where $\hat{v_{i}}$ indicates that vertex $v_{i}$ has been deleted from the simplex. For brevity, we use $[\hat{v_{i}}]$ to denote the simplex $[v_{0},v_{1},\ldots,v_{i-1},\hat{v_{i}},v_{i+1},\ldots,v_{n}]$ in computations.

Since the boundary operator can only be applied to oriented chains ($[\Delta^{n}]$ and $[P]$), and not graphs ($\Delta^{n}$ and $G$), we omit the square brackets for brevity.

Note that given a prodsimplicial cell complex $\Gamma$, $\partial_{N}$ defines a group homomorphism $\partial_{N}:C_{N}(\Gamma)\longrightarrow C_{N-1}(\Gamma)$. We verify that the above defined operator indeed defines a chain complex. Recall, for a simplex $\Delta^{n+1}$ with vertices $\{v_{0},v_{1},\ldots,v_{n+1}\}$, we have $(\partial_{n}\circ\partial_{n+1})(\Delta^{n+1})=0$.

With this boundary operator we can compute cycle groups, boundary groups, homology groups, and Betti numbers for prodsimplicial complexes of directed graphs.

Figure 4 shows two connected digraphs on three vertices. The one in Figure 4(a) is a closed path of the form $(v_{0},v_{1},v_{2}=v_{0})$ and forms a 1-cycle, while simplicial digraphs, as shown in Figure 4(b) do not contribute to $\beta_{1}$. Although closed paths of the form $(v_{0},v_{1},\ldots,v_{n}=v_{0})$ contribute to $\beta_{1}$, they are absent in consistently directed graphs, so we exclude them in the search for homology group generators.

The only two consistently directed graphs on four vertices are shown in Figure 5. One is the union of an edge and a path of length three, as depicted in Figure 5(a), which results in a complex homeomorphic to the circle $S^{1}$. Another is a square cell of the form $\Delta^{1}\square\Delta^{1}$, shown in Figure 5(b). We summarize our observations as follows.

We can build graphs with arbitrarily large $\beta_{1}$ values by attaching edge-disjoint paths of length 3 from source to target as follows.

This section will focus on directed graphs with vertices labeled $v_{i}$, so we reserve the notation $[v_{0},v_{1},\ldots,v_{n}]$ for simplices and label other low-dimensional prodsimplicial cells as sequences of vertices. For example, $[v_{0},v_{1},v_{2},v_{3}]$ denotes a tetrahedron while $v_{0}v_{1}v_{2}v_{3}$ denotes a square.

In dimension 2, the problem of finding all graphs whose corresponding prodsimplicial complex yields nontrivial 2-cycles is less straightforward. We present here a few of the simplest examples.

To consider generators of the second homology groups, we consider consistently directed graphs of at least five vertices, as Betti numbers for all consistently directed graphs on three and four vertices as shown in Figures 6 and 5. It can be checked by inspection that the only connected and consistently directed graphs on five edges and five vertices are pairs of paths as in Lemma 3.3. For an example with six edges and five vertices with $\beta_{2}(G)=1$, we consider the graph consisting of three squares as shown in Figure 7.

In addition to this, we present another graph on five vertices that has nontrivial second homology group that also yields a 2-cycle.

Using some of the graphs from previous results, we construct graphs with larger order having specific homology groups. To prove this result we use the following lemma that follows from the Mayer-Vietoris Sequence [17].

As a direct result of this, we are able to “glue” together an arbitrary number of generator graphs to attain any combination of Betti numbers.

It is therefore possible to construct consistently directed graphs with first and second homology groups isomorphic to $\mathbb{Z}^{k}$ and $\mathbb{Z}^{\ell}$, respectively, for any positive integers $k$ and $\ell$. This construction is not minimal on the number of vertices. For example, the graph shown in Figure 10 follows a construction similar to that in Lemma 3.2 and achieves many of the same $k$ values using fewer vertices and edges.

We call an ordered, countable set of symbols $\Sigma$ an alphabet. A word over $\Sigma$ is a finite sequence of the form $w=a_{1}a_{2}\ldots a_{n}$ where $a_{i}\in\Sigma$ whose length, denoted $|w|$, is $n$. We denote with $\Sigma^{*}$ the set of all words over $\Sigma$, including the empty word, denoted by $\epsilon$. The set of all symbols comprising a word $w$ is denoted by $\Sigma[w]$. The reverse of a word $w=a_{1}a_{2}\ldots a_{n}$ is $w^{R}=a_{n}a_{n-1}\ldots a_{2}a_{1}$. The word $v$ is a factor of the word $w$, denoted $v\sqsubseteq w$, if there exist $w_{1},w_{2}\in\Sigma^{\ast}$ such that $w=w_{1}vw_{2}$. In this presentation we set $\Sigma\subseteq[n]$ for some $n\in\mathbb{N}$. For example $w=122313$ is a word over $[3]=\{1,2,3\}$ of length $\lvert w\rvert=6$. The reverse of $w=122313$ is the word $w^{R}=313221$.

A word $w\in\Sigma^{*}$ is called a double occurrence word (DOW) if every symbol in $\Sigma$ appears in $w$ either zero or two times. We use $\Sigma_{DOW}$ to denote the set of all DOWs over $\Sigma$. Similarly, we call $w$ a single occurrence word (SOW) if each symbol in $\Sigma$ appears either once or not at all. The set of SOWs over $\Sigma$ is denoted by $\Sigma_{SOW}$. Since a DOW of length $n$ uses $n/2$ distinct symbols, we say that the size of a DOW $w$ is $|w|/2$. When we restrict $\Sigma_{DOW}$ to DOWs of size less than or equal to $n$, we denote the set by $\Sigma_{DOW}^{\leq n}$.

A word $w\in\Sigma^{*}$ is said to be in ascending order if $a_{1}=\min(\Sigma[w])$ and the first appearance of each symbol is the immediate successor of the largest of all the preceding symbols. For example, the word $w_{1}=122313$ is a DOW in ascending order, while the DOW $w_{2}=133212$ is not.

We say that $w_{1}$ and $w_{2}$ are ascending order equivalent, and write $w_{1}\sim w_{2}$, if there exists a bijection on $\Sigma$ inducing a morphism $f$ on $\Sigma^{\ast}$ such that $f(w_{1})=w_{2}$. Words $w_{1}=122313$ and $w_{2}=133212$ are equivalent via the bijective map given by: $1\mapsto 1,\;3\mapsto 2,\;2\mapsto 3.$ Since words in ascending order are unique, up to this equivalence we consider words in ascending order as representatives of the classes determined by the relation $\sim$.

From now on, we consider only equivalence classes of DOWs and abuse the notation by writing words in place of their equivalence class where no confusion arises. The following definition can be found in [9].