Construction numbers: How to build a graph?

TL;DR

This paper investigates the construction sequences of graphs, defining the construction number as the count of valid sequences, and computes this number for various graph families to understand their structural properties.

Contribution

It introduces the concept of construction numbers for graphs and provides formulas for calculating them across different graph families, advancing graph enumeration methods.

Findings

Determined construction numbers for several graph families.

Established formulas for counting valid construction sequences.

Enhanced understanding of graph structural enumeration.

Abstract

A construction sequence for a graph is a listing of the elements of the graph (the set of vertices and edges) such that each edge follows both its endpoints. The construction number of the graph is the number of such sequences. We determine this number for various graph families.

Equations

$$x^{-1}(s):=j\in [k]⟺x_j=s.$$

$$\ell :=\ell (G):=p+q$$

$$\forall e\in E,e=uw\Rightarrow x^{-1}(e)>\max (x^{-1}(u),x^{-1}(w)).$$

$$\mathrm{Perm}(V)\times \mathrm{Perm}(E)⊊\mathcal{C}(G)⊊\mathrm{Perm}(V\cup E)=\mathrm{Perm}(S_G).$$

$$p!q!\le c(G)\le (p+q)!$$

$$x^{\prime}=\Big{(}x^{-1}|S_{G^{\prime}}\Big{)}^{-1}\;\circ\;\zeta,$$

$$y\in \mathcal{C}(G)⟺\forall e=vw\in E(G)\setminus E(G^{\prime }),y^{-1}(e)>\max (y^{-1}(v),y^{-1}(w)).$$

$$312ab,312ba,132ab,132ba,123ab,123ba,12a3b,$$

$$321ab,321ba,32b1a,231ab,231ba,23b1a,213ab,213ba,21a3b.$$

$$c(G_1\cup G_2)=c(G_1)c(G_2)\left(\genfrac{}{}{0ex}{}{{\ell }_1+{\ell }_2}{{\ell }_1}\right),$$

$$c(G)=\prod _{i=1}^{n}c(G_i)\left(\genfrac{}{}{0ex}{}{\ell }{{\ell }_1\cdots {\ell }_n}\right).$$

$$\mathcal{C}(v,G):=\{x\in \mathcal{C}(G):v=x_1\}\text{mbox}and\mathcal{C}(v,w,G):=\{x\in \mathcal{C}(G):v=x_1,w=x_2\}$$

$$c(G)=\sum _{v\in V}c(v,G)=\sum _{(v,w)\in V\times V}c(v,w,G).$$

$$(G,v):=\underset{i=1}{\overset{n}{\bigvee }}(G_i,v_i):=\frac{{\coprod }_{i\in [n]}{G}_i}{v_1\equiv v_2\equiv \cdots \equiv v_n\equiv v}.$$

$$c(v,G)=\prod _{i=1}^{n}c(v_i,G_i)\left(\genfrac{}{}{0ex}{}{\ell -n}{{\ell }_1-1\cdots {\ell }_n-1}\right).$$

$$\mathcal{C}(G,e)=\{y\ast e:y\in \mathcal{C}(G-e)\}$$

$$c(G)=\sum _{e\in E}c(G-e).$$

$$x_{(i)}:=(x_1,\dots ,x_i)\in \mathcal{C}(G_i).$$

$$\Big{\{}(v_{1},v_{2},e_{1,2},v_{3})\Big{\}}\cup\Big{\{}(v_{1},v_{2},v_{3},e_{i,j})\Big{|}1\leq i<j\leq 3\Big{\}}\cup\Big{\{}(v_{1},v_{2},v_{3},v_{4})\Big{\}},$$

$$\frac{c(K_{1,n})}{(n+1)!n!}=\frac{2^n(n!{)}^2}{(n+1)!n!}=\frac{2^n}{n}$$

$$\frac{(2n+1)!}{c(K_{1,n})}=\frac{(2n+1)!}{2^n(n!{)}^2}\sim \frac{2^{1+n}\sqrt{n}}{\sqrt{\pi }}=c{2}^n\sqrt{n},\text{mbox}forc=\frac{2}{\sqrt{\pi }}\approx \mathrm{1.128....}$$

$$c(D_{a,b})=f(a,b)+\beta (a,b)c(D_{a-1,b})+\beta (b,a)c(D_{a,b-1}),$$

$$c(P_p)=T_p=(1/p)\left(\genfrac{}{}{0ex}{}{2^{2p}}{2}\right)|B_{2p}|.$$

$$\frac{c(P_{p})}{c(K_{1,p-1})}\sim\Big{(}4e^{-2}\ \pi^{-1/2}\Big{)}p^{1/2}\Big{(}\frac{8}{\pi^{2}}\Big{)}^{p}.$$

$$c(C_p)=\left(\genfrac{}{}{0ex}{}{2^{2p}}{2}\right)|B_{2p}|.$$

$$c(v,P_n)=S_n,$$

$$\nu (e,x):=2x^{-1}(e)-x^{-1}(u)-x^{-1}(w).$$

$$\nu (x):=\sum _{e\in E(G)}\nu (e,x)$$

$$y:=\mathbb{G}(\omega )$$

$$a(n+1)=na(n)⟺n+1\text{mbox}isprime.$$

Full text

Key Words: Roller-coaster problem, up-down permutations, minimum edge delay.

1 Introduction

The elements of a graph $G=(V,E)$ are the set $V\cup E$ of vertices and edges. A linear order on $V\cup E$ is a construction sequence (or c-sequence) for $G$ if each edge appears after both of its endpoints. For instance, for the path $P_{3}$ with vertices $1,2,3$ and edges $12,23$, one has construction sequences $(1,2,3,23,13)$ and $(1,2,12,3,23)$, while $(1,3,12,2,23)$ is not a c-sequence. There are a total of 16 c-sequences for $P_{3}$.

From a more sophisticated viewpoint, a construction sequence is a linear extension of the poset determined by the elements of the graph under inclusion, and the idea of counting these linear extensions goes back to Stanley [12]. More on this below.

Let ${\cal C}(G)$ be the set of all c-sequences for $G=(V,E)$. The construction number of $G$ is $c(G):=\#{\cal C}(G)$, the number of distinct construction sequences. So $c(P_{3})=16$.

Construction numbers (c-numbers) grow rapidly with the number of elements; finding these numbers for various graph families leads to interesting integer sequences (see the Monthly problem [8]). Here, we determine such sequences for various parameterized families of graphs (paths, stars, and complete graphs). But corresponding sequences for hypercubes and the multipartite graphs do not appear to be known.

Construction number behaves nicely with respect to disjoint union (Lemma 3 below); its behavior for other graph operations is unknown.

A natural question is: How do the density and topological connectivity of a graph affect construction number? For example, for trees with a fixed number of nodes, paths have the lowest and stars have the highest construction number [9], but corresponding problems are open for maximal planar and maximal outerplanar graphs.

We give a few simple properties of the construction sequence which allow vertex- and edge-based recursions. In particular, the first two terms are vertices and the last $k$ terms are edges, where $k$ is at least the minimum degree of the graph. A c-sequence can be enriched by replacing the $i$-th term by the subgraph determined by the elements in the first $i$ terms. Such a graphical c-sequence can be constrained by connectivity and other properties. For instance, a connected nontrivial graph has a construction sequence where each subgraph has at most two connected components.

Furthermore, the family of c-sequences of fixed length $n$ (arising from all graphs) is of a finite set of types. This is essentially the grammar of construction as a language where “vertex” and “edge” are the words. From the numerical evidence, it appears likely that the set of these construction types is equinumerous with a special family of rooted, decorated trees. As we sketch below, for $n=1,\ldots,6$, there are $1,1,2,5,19,87$ c-types, and these same numbers appear in [11, A124348].

An “opportunity cost” is introduced which charges each edge for the number of steps in the c-sequence by which it follows the placements of both its endpoints. We consider min- and max-cost c-sequences and a greedy algorithm that could allow graphical structure to be built up according to various rubrics.

Section 2 has basic definitions, Section 3 studies construction sequences for graphs with basepoint or basepoint-pairs, and Section 4 considers variability and construction type. In Section 5, we find $c(G)$ for various families. Section 6 defines cost-functions, min- and max-cost c-sequences, and relative constructability of graphs. Section 7 describes earlier appearances of construction numbers (notably [12], [14]) and shows how to extend c-sequences for hypergraphs, simplicial complexes, CW-complexes, posets, and categories. The last section has problems and applications.

2 Basic definitions and lemmas

Let $P_{p}$ be the path with $p$ vertices, $K_{1,p-1}$ the star with a single degree-$p{-}1$ hub having $p{-}1$ neighbors, each an endpoint; $C_{p}$ and $K_{p}$ are the cycle graph and complete graph with $p$ vertices. See, e.g., Harary [7] for any undefined graph terminology.

For any set $S$, $\#S$ is the cardinality, always finite in this paper. For $G=(V,E)$ a graph, we write $p=|G|=\#V$ for the order of $G$ and $q=\|G\|=\#E$ for the size. We use $n$ for the number of connected components and $m$ for cycle rank. Recall that $m=q-p+n$, so $m$ is the cardinality of a cycle basis. The maximum and minimum vertex-degree of $G$ are $\Delta$ and $\delta$. We write $[k]$ for $\{1,\ldots,k\}$, $k$ a positive integer. Let $\mathbb{N}_{>0}$ be the set of positive integers.

If $S$ is a finite set with $\ell$ elements, ${\rm Perm}(S)$ denotes the set of all bijections $x$ from $[\ell]$ to $S$, where $x:i\mapsto x_{i}$ for $i\in[\ell]$. A permutation on $S$ is an element of ${\rm Perm}(S)$. For $s\in S$ and $x=(x_{1},\ldots,x_{\ell})\in{\rm Perm}(S)$, we have

[TABLE]

If $G=(V,E)$ is a graph, we put $S_{G}:=V\cup E$, the set of elements of $G$ and let

[TABLE]

A permutation $x$ in ${\rm Perm}(S_{G})$ is a construction sequence (c-sequence) for the graph $G=(V,E)$ if each edge follows both its endpoints; i.e., if

[TABLE]

Let ${\cal C}(G)$ be the set of all c-sequences for $G$. Then for $G=(V,E)$ non-edgeless

[TABLE]

Any c-sequence in the subset ${\rm Perm}(V)\times{\rm Perm}(E)$ is called an easy c-sequence. The other construction sequences are called non-easy.

Let $c(G):=\#{\cal C}(G)$ be the construction number of $G$. For each graph $G$,

[TABLE]

We call these the vertex-edge bounds on the construction number, and by the containments in (4), the inequalities are strict if $\|G\|\geq 1$.

Lemma 1**.**

If $G$ is connected, then the last $\delta(G)$ entries in any construction sequence $x\in{\cal C}(G)$ are edges. Moreover, for each $v\in V(G)$, $x^{-1}(v)\leq\ell-\deg(v,G)$.

The first statement follows from the second, and the second from the definitions.

If $G^{\prime}\subseteq G$ and $x\in{\cal C}(G)$, then we define $x^{\prime}:=x\,|_{G^{\prime}}$, the restriction of $x$ to $G^{\prime}$, by

[TABLE]

where $\zeta$ is the order-preserving bijection from $[\ell^{\prime}]:=[\ell(G^{\prime})]$ to the subset of $[\ell]$ determined by the positions of the elements of $G^{\prime}$ in the sequence $x$.

Lemma 2**.**

Let $G^{\prime}\subseteq G$. If $x\in{\cal C}(G)$, then $x^{\prime}:=x|_{G^{\prime}}\in{\cal C}(G^{\prime})$. If $y$ is a permutation on $S_{G}$ such that $y^{\prime}:=y|_{G^{\prime}}\in{\cal C}(G^{\prime})$, then

[TABLE]

The number of ways to extend a c-sequence for $G-v$ to a c-sequence for $G$ depends on the particular construction sequence. For example, take $P_{2}=(\{1,2\},\{a\})$, where $a$ is the edge $12$. Then ${\cal C}(P_{2})=\{x^{\prime},y^{\prime}\}$, where $x^{\prime}:=(1,2,a)\equiv 12a$ and $y^{\prime}\equiv 21a$ (in short form). Consider $P_{3}=(\{1,2,3\},\{a,b\})$, where $b=23$. As $P_{2}\subset P_{3}$, each c-sequence for $P_{3}$ extends a c-sequence of $P_{2}$. One finds that $x^{\prime}$ has exactly 7 extensions to an element in ${\cal C}(P_{3})$ (only one of which is non-easy)

[TABLE]

while $y^{\prime}$ has exactly 9 extensions (two of which are non-easy)

[TABLE]

These are the 16 elements of ${\cal C}(P_{3})$.

Given two element-wise disjoint finite sequences $s_{1}$ and $s_{2}$ of lengths $n$ and $m$, we define a shuffle of the two sequences to be a sequence of length $n+m$ which contains both $s_{1}$ and $s_{2}$ as subsequences. The number of shuffle sequences of $s_{1}$ and $s_{2}$ is ${{n+m}\choose{n}}$, giving the construction number of a disjoint union in terms of its parts.

Lemma 3**.**

If $x_{1}$ and $x_{2}$ are c-sequences for disjoint graphs $G_{1}$ and $G_{2}$, resp., then each shuffle of $x_{1}$ and $x_{2}$ is a c-sequence for $G_{1}\cup G_{2}$, and conversely; hence,

[TABLE]

where $\ell_{1}$ and $\ell_{2}$ are the lengths of the sequences $x_{1}$ and $x_{2}$, resp.

The previous lemma extends to any finite pairwise-disjoint union of graphs. If $G_{1},\ldots,G_{n}$ have $\ell_{1},\ldots,\ell_{n}$ elements, then for $\ell:=\ell_{1}+\cdots+\ell_{n}$, replacing binomial by multinomial coeffient, the pairwise-disjoint union (coproduct) $G:=\coprod_{i=1}^{n}G_{i}$ satisfies

[TABLE]

Construction numbers are equal for isomorphic graphs. More exactly, we have

Lemma 4**.**

Let $\phi:G\to H$ be an isomorphism of graphs. Then there is an induced bijection $\hat{\phi}:{\cal C}(G)\to{\cal C}(H)$ given by $\hat{\phi}(x)=\hat{x}$, where $\hat{x}_{j}=\phi(x_{j})$ for $1\leq j\leq\ell(G)$.

3 Construction sequences for basepointed graphs

Let $H$ be any subgraph of $G$ and let $y$ be any construction sequence of $H$. Then the set of all construction sequences of $G$ which

A graph with basepoint is an ordered pair $(G,v)$, where $v\in V(G)$. A graph with basepoint-pair is an ordered triple $(G,v,w)$, where $(v,w)$ is an ordered pair of distinct vertices in $V(G)$. We call $v$ a basepoint and $(v,w)$ a basepoint-pair. Let

[TABLE]

be the sets of construction sequences for the corresponding graphs beginning with the basepoint or basepoint-pair. The based construction number $c(v,G):=\#{\cal C}(v,G)$ is the number of c-sequences starting with $v$. The pair-based construction number $c(v,w,G):=\#{\cal C}(v,w,G)$ is defined analogously. By definition, ${\cal C}(v,w,G)$ is empty if $v=w$.

As each c-sequence starts with a pair of vertices, we have the following expressions for the construction number of a graph as the sum of its based construction numbers,

[TABLE]

The based construction numbers behave nicely w.r.t. isomorphism.

Lemma 5**.**

Let $G=(V,E)$, $G^{\prime}=(V^{\prime},E^{\prime})$ be graphs. If $v\in V$, $v^{\prime}\in V^{\prime}$, and if $\phi:G\to G^{\prime}$ is an isomorphism such that $\phi(v)=v^{\prime}$, then $c(v,G)=c(v^{\prime},G^{\prime})$.

Proof.

Let $x\in{\cal C}(G)$. Then $x^{\prime}:=(\phi(x_{1}),\ldots,\phi(x_{\ell}))$ is a c-sequence for $G^{\prime}$. ∎

Similarly, pair-based construction sequences behave well w.r.t. isomorphism.

Lemma 6**.**

Let $G=(V,E)$, $G^{\prime}=(V^{\prime},E^{\prime})$ be graphs. If $v,w\in V$, $v^{\prime},w^{\prime}\in V^{\prime}$, and if $\phi:G\to G^{\prime}$ is an isomorphism such that $\phi(v)=v^{\prime}$ and $\phi(w)=w^{\prime}$, then $c(v,w,G)=c(v^{\prime},w^{\prime},G^{\prime})$.

A graph $G$ is vertex-transitive if for every pair $(v,w)$ of vertices, there is an automorphism of $G$ carrying $v$ to $w$. The based construction number of a vertex-transitive graph is independent of basepoint. Hence, by (9) and Lemma 5, we have

Proposition 1**.**

If $G=(V,E)$ is vertex transitive, then $c(G)=p\cdot c(v,G)$.

For $i=1,\ldots,n$, let $G_{i}$ be a family of graphs with $\ell_{i}$ elements and chosen basepoints $v_{i}\in VG_{i}$. We now define a coproduct for graphs with basepoint which is the disjoint union of the $G_{i}$ with basepoints $v_{i}$ all identified to a single vertex $v$, also called the “wedge-product”

[TABLE]

be the resulting graph which has base point $v$. Then as in (8), we have

[TABLE]

where $\ell:=\ell_{1}+\cdots+\ell_{n}$.

We can also give an edge-based recursion. Let $G=(V,E)$ be nontrivial connected and for $e\in E$ write ${\cal C}(G,e):=\{x\in{\cal C}(G):x_{\ell}=e\}$; put $c(G,e):=\#{\cal C}(G,e)$.

Lemma 7**.**

Let $G=(V,E)$ be nontrivial connected. Then

[TABLE]

and

[TABLE]

Call a graph edge transitive if there is an automorphism carrying any edge to any other edge. Then $c(G,e)$ is independent of the edge.

Proposition 2**.**

Let $G=(V,E)$ be edge transitive. Then $c(G)=q\cdot c(G,e)$.

4 Structure in c-sequences

In this section, we consider a construction sequence as a message about the unknown graph it is building and ask: How many length-$n$ messages are possible? It amounts to the richness of the grammar, namely the number $\rho(n)$ of distinct graphs that can be encoded by a c-sequence of length $n$. We shall see that the family of all $\rho(n)$ messages can be partitioned into a much smaller set of $\tau(n)$ types.

Let $G=(V,E)$ be a graph and let $x\in{\cal C}(G)$. The graphical sequence of $x$ is the sequence $\{G_{i}(x)\}$, $1\leq i\leq\ell$, of subgraphs of $G$, where $G_{i}:=G_{i}(x):=G(x_{(i)})$ and

[TABLE]

For any graph $H$, let $n(H)$ denote the number of connected components and let $m(H)$ denote the cycle rank, so $m(H)=q(H)-p(H)+n(H)$ [7, p 39].

Lemma 8**.**

If $G$ is a graph, $2\leq i\leq\ell(G)$, and $x\in{\cal C}(G)$, one of the following occurs:

(i) $n(G_{i}(x))-n(G_{i-1}(x))=1$, $m(G_{i}(x))-m(G_{i-1}(x))=0$,

(ii) $n(G_{i}(x))-n(G_{i-1}(x))=0$, $m(G_{i}(x))-m(G_{i-1}(x))=1$, or

(iii) $n(G_{i}(x))-n(G_{i-1}(x))=-1$, $m(G_{i}(x))-m(G_{i-1}(x))=0$,

according to whether $x_{i}$ is a vertex, an edge joining two vertices in the same connected component of $G_{i-1}$, or an edge joining two vertices in different components of $G_{i-1}$.

One can describe the graph of a construction sequence by keeping track, when an edge is added, which two of the existing vertices are being joined or noting that a vertex is being added. We write $\rho(n)$ for the number of distinct c-sequence with $n$ elements that can arise for any possible graph, connected or not.

Putting $e_{i,j}:=v_{i}v_{j}$, the set of construction sequences of length 4 is

[TABLE]

so $\rho(4)=5$. That $\rho(1)=\rho(2)=1$ and $\rho(3)=2$ are easily checked. Also, $\rho(5)=19$.

Indeed, for $n=5$, one has the following construction sequences in 6 types:

$v_{1},v_{2},e_{12},v_{3},v_{4}$;

$v_{1},v_{2},e_{12},v_{3},e_{j,3}$,   for $\;1\leq j\leq 2$;

$v_{1},v_{2},v_{3},e_{i,j},e_{r,s}$,   for $\;e_{i,j}\in E(T(1,2,3))$, $e_{r,s}\in E(T(1,2,3)-ij)$;

$v_{1},v_{2},v_{3},e_{i,j},v_{4}$,   for $\;e_{i,j}\in E(T(1,2,3))$;

$v_{1},v_{2},v_{3},v_{4},e_{i,j}$   for $1\leq i<j\leq 4$;

$v_{1},v_{2},v_{3},v_{4},v_{5}$;

where $T(1,2,3)$ denotes the $K_{3}$ graph determined by vertices $v_{1},v_{2},v_{3}$. Actually, we could simplify notation by dropping subscripts. So the first type is $vvevv$ and the last $vvvvv$. The 6 types contribute, respectively, $1+2+6+3+6+1=19$ c-sequences, so $\rho(5)=19$ as claimed. We check that $\rho(6)=87$ in the Appendix.

The sequence $(1,1,2,5,19,87)$ agrees with $A124348$ in [11], and we ask if there exists a 1-1 correspondence between the set of c-sequences with $n$ elements and the set of “increasing rooted trees on $n$ nodes with thinning limbs” enumerated by [11, A124348]. The putative one-to-one correspondence is non-obvious.

A rooted tree is increasing if it is labeled so that labels increase along any path away from the root. A rooted tree has thinning limbs if the number of children of each vertex $v$ is not greater than the number of children of $v$’s parent, its predecessor on the unique path from the root. There are a total of 6 distinct rooted unlabeled trees of order 5 with thinning limbs, which does correspond to the 6 types.

Listing these 6 unlabeled trees in reverse lexicographic order according to the sequence of heights from root to leaves, they have the following numbers of increasing labelings: $1,4,4,3,6,1$, making 19 in all. The two partitions of 19 are distinct though they have the same number of parts. But our analysis for $n=6$ in the Appendix finds 12 types, rather than 10 which is the number of distinct rooted unlabeled trees of order 6 with thinning limbs [11, A124343].

Finally, we include a few remarks on the possibility of including conditions on the partial graphs $G_{i}$. A real sequence $(a_{1},\ldots,a_{r})$ is monotonically increasing if $a_{1}\leq\cdots\leq a_{r}$. Clearly, for any $x\in{\cal C}(G)$, the sequence $(m(G_{1}(x)),\ldots,m(G_{\ell}(x))$ is monotonically increasing.

We call a real sequence $(a_{1},\ldots,a_{r})$ strongly left unimodal (SLU) if there is a (unique) index $k\in[r]$ such that $a_{1}<a_{2}<\ldots<a_{k}$, and $a_{k}\geq a_{k+1}\geq\cdots\geq a_{r}$.

A c-sequence is easy if all its vertices are consecutive; see equation (4). Proofs of the two results below are left to the reader.

Proposition 3**.**

*If $G$ has order $p$ and $x\in{\cal C}(G)$, then the following are equivalent:

(1) $(n(G_{1}(x)),\ldots,n(G_{\ell}(x))$ is SLU, where $\ell:=\ell(G)$);

(2) $n(G_{p}(x))=p$;

(3) $x$ is easy.*

Let $G$ be a graph and $x\in{\cal C}(G)$. We define $n_{\max}(x):=\max_{1\leq i\leq\ell(G)}n(G_{i}(x))$ and $n_{\max}(G):=\min_{x\in{\cal C}(G)}n_{\max}(x)$. A construction sequence $x$ for $G$ is called connected if $n_{\max}(x)=n_{\max}(G)=2$. Then we have

Proposition 4**.**

*For any nontrivial graph $G$, the following are equivalent:

(1) $n_{\max}(G)=2$;

(2) $G$ is connected;

(3) $G$ has a connected c-sequence.*

5 Construction numbers for some graph families

In this section, construction numbers are determined for stars and double-stars, for paths and cycles, and for complete graphs. We begin with an easy example.

Theorem 1**.**

For $n\geq 0$, $c(K_{1,n})=2^{n}(n!)^{2}$.

Proof.

For $n=0,1$, the result holds. Suppose $n\geq 2$ and let $x=(x_{1},\ldots,x_{2n+1})$ be a construction sequence for $K_{1,n}$. There are $n$ edges $e_{i}=v_{0}v_{i}$, where $v_{0}$ is the central node, and one of the edges, say, $e_{i}$, must be the last term in $x$. This leaves $2n$ coordinates in $x^{\prime}:=(x_{1},\ldots,x_{2n})$ and one of them is $v_{i}$. The remaining $(2n-1)$ coordinates are a construction sequence for the $(n-1)$-star $K_{1,n-1}$. Hence, $c(K_{1,n})=n(2n)c(K_{1,n}-v_{i})=2n^{2}2^{n-1}(n-1)!^{2}=2^{n}(n!)^{2}$ by induction. ∎

The numbers 2, 16, 288, 9216, 460800 generated by the above formula count the number of c-sequences for $K_{1,n}$ for $n\in\{1,2,3,4,5\}$. These numbers are the absolute value of the sequence A055546 in the OEIS (reference [11]) and describe the number of ways to seat $n$ cats and $n$ dogs in a roller coaster with $n$ rows, where each row has two seats which must be occupied by a cat and a dog.

Interestingly, the value of $c(K_{1,n})$ asymptotically agrees with the geometric mean of the vertex-edge bounds (5) up to a constant multiple of $n^{3/2}$. For we have

[TABLE]

while

[TABLE]

A double star is a tree of diameter at most 3. For $a,b\geq 0$, let $D_{a,b}$ denote the double star formed by the union of $K_{1,a+1}$ and $K_{1,b+1}$ with one common edge. So $D_{a,b}$ consists of two adjacent vertices, say $v$ and $w$, where $v$ is adjacent to $a$ and $w$ to $b$ additional vertices, and all vertices other than $v$ and $w$ have degree 1.

Using the recursion (11) as well as equation (7), one gets the following.

Theorem 2**.**

Let $a,b\geq 0$ be integers. Then

[TABLE]

where $f(a,b)=c(K_{1,a}\coprod K_{1,b})={{2a+2b+2}\choose{2a+1}}2^{a}(a!)^{2}2^{b}(b!)^{2}$, and $\beta(a,b):=a(2a+2b+2)$.

We don’t yet have a closed form but can calculate the values. For example, $D_{2,2}$, the $6$ vertex tree with two adjacent cubic nodes, has 402432 c-sequences. But $c(K_{1,5})=460800$. In fact, stars maximize the construction number over all trees of a fixed order. In [9], the following is shown.

Lemma 9**.**

If $T$ is any $p$-vertex tree, then $c(P_{p})\leq c(T)\leq c(K_{1,p-1})$.

Hence, the extremal trees have extremal construction numbers.

For the path, we show below that $c(P_{p})$ is the $p$-th Tangent number $T_{p}$. By [18], [4, 24.15.4]), where $B_{2p}$ is the $2p$-th Bernoulli number [11, (A027641$/$A027642)],

[TABLE]

An asymptotic analysis shows $c(P_{p})$ is exponentially small compared to $c(K_{1,p-1})$.

[TABLE]

By Proposition 2, counting c-sequences for $C_{p}$ and $P_{p}$ are equivalent problems.

Lemma 10**.**

If $p\geq 3$, then $c(C_{p})=p\cdot c(P_{p})$.

Using Lemma 10 and equation (15), we have for $p\geq 3$,

[TABLE]

In fact, the formula holds for $p\geq 1$, where cycles are CW-complexes: $C_{1}$ has one vertex and a loop, and $C_{2}$ has two vertices and two parallel edges.

Before determining $c(P_{p})$, we give a Catalan-like recursion for these numbers.

Lemma 11**.**

$c(P_{n})=\sum_{k=1}^{n-1}c(P_{k})\,c(P_{n-k})\,{{2n-2}\choose{2k-1}}.$**

Proof.

Any construction sequence $x$ for $P_{n}$ has last entry an edge $e$, whose removal creates subpaths with $k$ and $n-k$ vertices, resp., for some $k$, $1\leq k\leq n-1$. Now $x$ contains construction sequences for both subpaths which suffices by Lemma 3. ∎

Trivially, $c(P_{1})=1$. By the recursion, one gets $1,2,16,272,7936,353792$ for $c(P_{n})$, $n=1,\ldots,6$, empirically [11, A000182], the sequence of tangent numbers, $T_{n}$. The equation was first proved by Stanley [14]; we give two simple arguments.

Theorem 3**.**

For all $n\geq 1$, $c(P_{n})=T_{n}$.

Proof.

Let $U(n)$ be the set of up-down permutations of $[n]$, where successive differences switch sign, and the first is positive. It is well-known [19] that $\#U(2n-1)=T_{n}$.

Dan Ullman observed that there is a bijection from ${\cal C}(P_{n})$ to $U(2n-1)$. Indeed, a permutation $\pi$ of the consecutively labeled elements of a path is a construction sequence if and only if $\pi^{-1}$ is an up-down sequence. Indeed, for $j=1,\ldots,n-1$, $\pi^{-1}(2j)$ is the position in $\pi$ occupied by the $j$-th edge, while $\pi^{-1}(2j-1),\pi^{-1}(2j+1)$ correspond to the positions of the two vertices flanking the $2j$-th edge and so are smaller iff $\pi$ is a construction sequence. (For example, if we label the elements of $P_{5}$ from left to right $(1,2,3,4,5,6,7,8,9)$, where odd-numbers correspond to vertices and even numbers to edges, then $P_{5}$ has the c-sequence $\pi=(1,3,5,7,9,2,4,6,8)$ and $\pi^{-1}=(1,6,2,7,3,8,4,9,5)$ is up-down.) So the theorem is established. ∎

Here is a second proof. Let $J_{r}$ be the number of permutations of $\{0,1,\ldots,r+1\}$ which begin with ‘$1$’, end with ‘[math]’, and have consecutive differences which alternate in sign. Observe that $J_{2k}=0$ for $k\geq 1$ as the sequences counted by $J$ must begin with an up and end with a down and hence have an odd number of terms. These tremolo sequences were shown by Street [15] to be in one-to-one correspondence with “Joyce trees” and to satisfy the following recursion.

Proposition 5** (R. Street).**

For $r\geq 3$, $J_{r}=\sum_{m=0}^{r-1}{{r-1}\choose{m}}J_{m}J_{r-1-m}$.

Street then proved that $J_{2n-1}=T_{n}$. Hence, it suffices to show that $c(P_{n})=J_{2n-1}$. Indeed, $J_{1}=c(P_{1})$ and $J_{3}=c(P_{2})$. If you replace $J_{2r-1}$ by $c(P_{r})$ and $J_{2r}$ by zero and re-index, then the recursion in Proposition 5 becomes that of Lemma 11, so $c(P_{n})$ and $J_{2n-1}$ both satisfy the same recursion and initial conditions.

If $v$ is one of the endpoints of $P_{n}$, we can calculate $c(v,P_{n})$ for the first few values, getting (with some care for the last term) $1,1,5,61$ for $n=1,2,3,4$. In fact,

[TABLE]

where $S_{n}$ is the $n$-th secant number ([11, A000364]), counting the “zig” permutations.

5.1 Complete graphs

This section is redacted until October 31 when solutions to the Monthly problem [8] are due for submission.

6 Costs for construction sequences

In this section, we model delay in edge-placement as a cost. Suppose one has a set of distinct buildings and links between them. Assuming some notion of profit or useful functionality once a link is placed between two buildings, there is an opportunity cost for delay in building the link. In sentences, there is a cost related to remembering that the verb relates its previously placed subject and object.

Once its two endpoints $u$ and $w$ appear in a c-sequence $x$, the edge $e=uw$ can be placed, giving a delay of at least 3, but when a vertex of high degree is placed in the list, it may suddenly allow many possible edges to appear, so the delay for some of these edges can be large. We now formalize these ideas.

Let $G=(V,E)$ be a graph, let $x\in{\cal C}(G)$, and let $e=uw\in E$. The cost of edge $e$ with respect to construction sequence $x$ is defined to be (see (1) and (2))

[TABLE]

The cost of a construction sequence $x\in{\cal C}(G)$ is the sum of the costs of its edges

[TABLE]

We say $x$ is min-cost if $\nu(x)=\nu_{*}(G):=\min\{\nu(y):y\in{\cal C}(G)\}$ is the least cost to construct $G$. Similarly, $x$ is max-cost if $\nu(x)=\nu^{*}(G):=\max\{\nu(y):y\in{\cal C}(G)\}$.

Let ${\cal C}_{*}(G)$ be the set of min-cost c-sequences for $G$ and let $c_{*}(G)$ be its cardinality. Let ${\cal C}^{*}(G)$ be the set of max-cost c-sequences for a graph $G$ and let $c^{*}(G)=|{\cal C}^{*}(G)|$.

Bounds or estimates on $\nu_{*}(G)$ allow us to determine how near some given c-sequence is to min-cost, while $\nu^{*}(G)$ tells us worst-case. Further, in some applications, delay might be a good thing - e.g., if a resource is becoming cheaper with time. See [6] regarding applications. Here we give the basics.

Any construction sequence $x$ for $G=(V,E)$ induces a vertex permutation $x|V$ on the set $V$ of vertices. Conversely, given a vertex permutation $\omega$, we now define a greedy algorithm $\mathbb{G}$, which produces a construction sequence $x$ such that $x|V=\omega$ where $x$ has minimum cost among all $\omega$-inducing c-sequences for $G$, as follows.

Let $G=(V,E)$ be a graph, $x\in{\cal C}(G)$, and $i\in[\ell(G)]$; edge $e=uw$ is available for $x\in{\cal C}(G)$ at time $i$ if $x^{-1}(e)\geq i$ but $\max(x^{-1}(u),x^{-1}(w))<i$. Availability of edge $e$ at time $i$ for $x$ is equivalent to the assertion that $(x_{1},\ldots,x_{i-1})$ contains $u$ and $w$ but not $e$. Let $A:=A(G,x,i)$ be the set of all edges available for $x$ at time $i$.

Let $G=(V,E)$ be connected and non-edgeless, let $\omega=(v_{1},v_{2},\ldots,v_{p})$, and define

[TABLE]

to be the c-sequence obtained by setting $y_{j}=v_{j}$ for $j=1,\ldots,a$, where $a\geq 2$ is the first index for which $V_{a}:=\{v_{1},\ldots,v_{a}\}$ induces a non-edgeless subgraph $G(V_{a})$ of $G$, with edge-set $E(G(V_{a}))=A(G,y,a{+}1)$. The elements $y_{a+1},\ldots,y_{a+b}$ are the $b\geq 1$ edges in $G(V_{a})$ in the lexicographic order induced by $\omega$. Take the next vertices from $\omega$ beginning with $y_{a+b+1}=v_{a+1}$ until again one or more edges become available, etc. Call $y$ a greedy c-sequence for $G$; let ${\cal G}(G)\subset{\cal C}(G)$ be the set of greedy c-sequences.

The available set of edges can actually be taken in any order.

Lemma 12**.**

Cost of a c-sequence is independent of the ordering of consecutive edges.

Proof.

It suffices to check this for the interchange of two adjacent edges. One of the edges moves left and so has its cost reduced by 2 while the edge that moves to the right has its cost increased by 2; so the cost for the sequence remains constant. ∎

Call two c-sequences $x,y$ for $G$ cost equivalent if they have the same cost and edge-rearrangement equivalent if costs agree after permuting within sets of consecutive edges. For c-sequences, min-cost implies equivalence to greedy.

Theorem 4**.**

Let $x\in{\cal C}(G)$ where $G$ is connected and nontrivial. If $x$ is min-cost, then there exists $y\in{\cal G}(G)$ such that $x$ and $y$ are edge-rearrangement equivalent.

Proof.

To minimize cost of a c-sequence $y$ which extends $(x_{1},\ldots,x_{i-1})$, all edges in $A=A(G,x,i)$ should be placed before any new vertices. Indeed, if vertex $v\notin V_{i-1}$ precedes edge $e$ in $A$, reversing the positions of $v$ and $e$ reduces the cost for any edge with $v$ as an endpoint while decreasing the cost for $e$ as well. ∎

The following simple result says that a max-cost c-sequence is also determined by the vertex order of the graph. “Easy” means all vertices first followed by all edges.

Theorem 5**.**

Let $x\in{\cal C}(G)$. If $x$ is max-cost, then $x$ is an easy sequence.

Proof.

If $x$ is not easy, some edge immediately precedes a vertex. Reversing them increases cost. ∎

Cost and applications are studied in [6], continuous or probabilistic models in [9].

Choice of cost function may be influenced by application. One might wish to concentrate resources, filling in a portion of the graph with all possible edges between the adjacent vertex-pairs so as to maximize the number of partial subgraphs that are induced subgraphs, hence reaching full density as early as possible. In contrast, the graph cost function might be designed to cause the induced subgraphs to become spanning as early as possible - e.g., by means of a spanning tree when the graph to be constructed is connected. A further goal, consistent with the spanning tree model, is to cause density to increase as slowly as possible which might be appropriate if high-volume vertex throughput requires a learning period. (Perhaps a suitable regularizing function for the cost would be quadratic in the vertex-degrees.)

7 Earlier appearances and extensions

Stanley studied the number of linear extensions of a partial order [12, p 8], using them to define a polynomial [13, p 130]. In [14, p 7] he showed the number of linear extensions of the partial order determined by a path is an Euler number, implying our results (15) and (17) above. (Stanley called such a poset a “fence”.)

Now take the Hasse diagram of any partial order; define a construction sequence for the diagram to be a total order on the elements of the poset such that each element is preceded in the linear order by all elements which precede it in the partial order. Simplicial and CW-complexes can be partially ordered by “is a face of”, and the linearization of posets includes graphs and hypergraphs as 2-layer Hasse diagrams.

Construction sequences make sense for hypergraphs, multigraphs, and indeed for any CW-complex, where each cell must follow the cells to which it is attached. One might start with simplexes, cubes, and hyperoctahedra (the standard Platonic polytopes), and the sporadic instances in 3 and 4 dimensions. Limits and colimits of diagrams in a category could be considered, even beyond the finite realm [12, p 77].

Other notions of graph construction are known. Whitney [20] has defined an ear decomposition. Suppose given a cycle; now attach nontrivial paths (“ears”) so that the endpoints of the path are the only intersection of the attached path with the union of the previous elements of the decomposition. Existence of an ear decomposition characterizes the 2-connected graphs. For minimum (or relatively low) cost c-sequences of a 2-connected graph, one might try using the ear decomposition as a guide since min-cost c-sequences are known for its parts.

For 3-connected graphs, Tutte has shown that all can be constructed starting from a wheel graph (the cone on a cycle) by splitting vertices and adding edges [7]. More generally, shellability and collapsability of complexes and matroids might also be admitted. But it isn’t clear how to exploit these structures for c-sequences.

A different approach to graph construction was motivated by the goal to describe the self-assembly of macromolecules performed by virus capsids in the host cell. An assembly tree [17] is a gadget that builds up a graph from subgraphs induced by various subsets of the vertices. The number of assembly trees of a graph is the assembly number but these seem to be quite unrelated to construction numbers. For example, for $n$-stars, [17] gives $n!$, while for paths and cycles, Vince and Boná find a Catalan-type value.

The set $[n]$ of positive integers under divisibility form a poset and the number $a(n)$ of linear extensions is enumerated in A016021 of [11]. We note that understanding this integer sequence would be quite interesting as

[TABLE]

8 Discussion

Aside from their combinatorial relevance for the structure of incidence algebras [12] or for enumeration and integer sequences [8, 9], construction numbers of graphs might have a deeper theoretical aspect as they do for the divisibility graph of $[n]$.

A natural idea is to think of construction sequences as the outcome of a constrained stochastic process, where a graph evolves through the addition of new vertices and edges subject to the condition that an edge cannot appear before its endpoints. Any given graph thus could be “enriched” by knowledge of its history, either the linear order on its elements or their actual time of appearance. The existence of such histories might enable some new methods of proof - e.g., for the graph reconstruction problem of Ulam and Harary.

Practical applications could include operations research, where directed hyperedges describe complex tasks such as “build an airbase” which depend on various supporting infrastructures. If a link should occur at some moment, one would like the necessary endpoints to happen just-in-time.

Graphs occur in many scientific contexts. It would be interesting to study the actual construction sequences for the complex biochemical networks found in the organic kingdom. How close are they to being min-cost?

Brightwell & Winkler [2] showed that counting the linear extensions of a poset is $\#P$-complete and contrast this with randomized polynomial-time algorithms which estimate this number. Their conjecture that $\#P$-completeness holds even for height-2 posets was proved by Dittmer & Pak [5]. Applications of linear extensions of posets to equidistributed classes of permutations were given by Björner & Wachs [2], and Burrow [3] has studied using traversals of posets representing taxonomies and concept lattices to construct algorithms for information databases.

Computer calculation of construction numbers to get sample values can aid in finding correct formulas (via the OEIS [11]) for inductive proofs, but such computation is difficult due to the large number of permutations. This might be flipped into an asset by utilizing the theoretical calculations here as a “teacher” for neural network or machine learning methods (cf. Talvitie et al. [16]). More ambitiously, a mathematically oriented artificial intelligence could be challenged to discover the formulas above, along with some of the others we would like to have.

Acknowledgement

We thank Stan Wagon for helpful conversations and for pointing out [17]. We also acknowledge Richard Hammack for noting we were implicitly using equation (11).

Appendix

Here is the calculation that $\rho(6)=87$ and $\tau(6)=12$. We find 12 types of c-sequence giving the indicated numbers of c-sequences which sum to 87:

$vvvvvv\;\;\;1\\ vvvvve\;\;\;10\\ vvvvev\;\;\;6\\ vvvvee\;\;\;30\\ vvvevv\;\;\;3\\ vvveve\;\;\;15\\ vvveev\;\;\;6\\ vvveee\;\;\;6\\ vvevvv\;\;\;1\\ vvevve\;\;\;5\\ vvevev\;\;\;2\\ vvevee\;\;\;2\\$