Daisy cubes: a characterization and a generalization

TL;DR

This paper characterizes daisy cubes, a class of isometric subgraphs of hypercubes, provides an expansion-based characterization, and introduces algorithms for embedding and generalizations.

Contribution

It offers a new characterization of daisy cubes via an expansion procedure and presents algorithms for embedding graphs into hypercubes.

Findings

Characterization of daisy cubes through expansion procedure

Algorithm for embedding graphs into hypercubes in O(mn) time

Introduction of daisy graphs as a generalization of daisy cubes

Abstract

Daisy cubes are a recently introduced class of isometric subgraphs of hypercubes $Q_n$. They are induced with intervals between chosen vertices of $Q_n$ and the vertex $0^n\in V(Q_n)$. In this paper we characterize daisy cubes in terms of an expansion procedure thus answering an open problem proposed by Klav\v{z}ar and Mollard, 2018, in the introductory paper of daisy cubes \cite{KlaMol-18}. To obtain such a characterization several interesting properties of daisy cubes are presented. For a given graph $G$ isomorphic to a daisy cube, but without the corresponding embedding into a hypercube, we present an algorithm which finds a proper embedding of $G$ into a hypercube in $O(mn)$ time. Finally, daisy graphs of a rooted graph are introduced and shown to be a generalization of daisy cubes.

Equations

$$Q_n(X)=⟨\{u\in B^n|u\le x\text{ for some }x\in X\}⟩.$$

$$G_r(X)=⟨\{u\in V(G)|u{\le }_{G,r}v\text{ for some }v\in X\}⟩.$$

Full text

Keywords. daisy cubes, characterization, generalization.

Mathematics Subject Classifications. 05C75, 05C85, 68R10.

1 Introduction

Daisy cubes have been recently introduced by Klavžar and Mollard [4]. This is a new class of partial cubes that contains other well known families of cube-like graphs, e.g. Fibonacci cubes ([3]) and Lucas cubes ([5, 6]). Since the introduction other results on daisy cubes have already appeared. Providing a connection to chemical graph theory, Žigert Pleteršek has shown in [9] that the resonance graphs of kinky benzenoid systems are daisy cubes. In [7] Vesel has shown that the cube-complement of a daisy cube is also a daisy cube.

In the introductory paper [4] several nice properties and open problems about daisy cubes are presented. Among others the following problem.

Problem 1.1**.**

[4, Problem 5.1]** Do daisy cubes admit a characterization in terms of an expansion procedure?

In this paper we show that the answer to Problem 1.1 is affirmative. We proceed as follows. In the continuation of this section we provide basic definitions and results needed throughout the paper. In section 2, in the process of providing a characterization of daisy cubes in terms of an expansion procedure, we prove several other interesting properties of daisy cubes. We continue with section 3 by providing an algorithm which for a given graph $G$ isomorphic to a daisy cube finds a proper labelling of the vertices of $G$ and thus an isometric embedding into the corresponding hypercube. The paper is concluded with a generalization of daisy cubes and some open problems.

Let $B=\{0,1\}$. A word $u$ of length $n$ over $B$ is called a binary string of length $n$ and will be denoted by $u=(u_{1},u_{2},\ldots,u_{n})\in B^{n}$, or shorter $u_{1}u_{2}\ldots u_{n}$. We will use the power notation for the concatenation of bits, for instance $0^{n}=0\ldots 0\in B^{n}$.

The $n$-cube $Q_{n}$ is the graph with the vertex set $B^{n}$, where two vertices are adjacent whenever the two binary strings differ in exactly one position.

For two vertices $u$ and $v$ of a graph $G$, the interval $I_{G}(u,v)$ between $u$ and $v$ in $G$ is the set of all vertices lying on some shortest $u,v$-path, that is, $I_{G}(u,v)=\{w\in V(G)\ |\ d(u,v)=d(u,w)+d(w,v)\}$. The index $G$ may be omitted where the graph will be clear from the context. A subgraph $H$ of a graph $G$ is isometric if $d_{H}(u,v)=d_{G}(u,v)$ holds for any $u,v\in V(H)$. Isometric graphs of hypercubes are called partial cubes.

A median of a triple of vertices $u$, $v$ and $w$ of a graph $G$ is a vertex $z\in V(G)$ that lies on a shortest $u,v$-path, on a shortest $u,w$-path, and on a shortest $v,w$-path. If $G$ is connected and every triple of vertices admits a unique median, then $G$ is a median graph. It is well known that median graphs are partial cubes (cf. [2]).

Let $G$ be a connected graph with $e=xy$ and $f=uv$ two edges in $G$. We say that $e$ is in relation $\Theta$ to $f$ if $d(x,u)+d(y,v)\not=d(x,v)+d(y,u)$. $\Theta$ is reflexive and symmetric, but need not be transitive. We denote its transitive closure by $\Theta^{*}$. It was proved in [8] that $G$ is a partial cube if and only if $G$ is bipartite and $\Theta=\Theta^{*}$.

For $X\subseteq V(G)$ we denote the subgraph of $G$ induced by the set $X$ with $\langle X\rangle_{G}$ or simply as $\langle X\rangle$, when $G$ is clear from the context.

A subgraph $H$ of a graph $G$ is called convex, if it is connected and if every shortest path of $G$ between two vertices of $H$ is completely contained in $H$.

For an edge $ab$ of graph $G$ we define:

• •

$W_{ab}=\{w\in V(G)\ |\ d(a,w)<d(b,w)\}$

• •

$W_{ba}=\{w\in V(G)\ |\ d(b,w)<d(a,w)\}$

• •

$F_{ab}=\{xy\in E(G)\ |\ x\in W_{ab}\text{ and }y\in W_{ba}\}$

• •

$U_{ab}=\{w\in W_{ab}\ |\ w\text{ is the end vertex of an edge in }F_{ab}\}$

• •

$U_{ba}=\{w\in W_{ba}\ |\ w\text{ is the end vertex of an edge in }F_{ab}\}$

Let $ab$ be an edge of a partial cube $G$ for which $U_{ab}=W_{ab}$. Then $\langle W_{ab}\rangle$ is called a peripheral subgraph of $G$. A $\Theta$-class $E$ of a median graph $G$ is called peripheral, if at least one of $\langle W_{ab}\rangle$ and $\langle W_{ba}\rangle$ is peripheral for $ab\in E$. $E$ is internal if it is not peripheral.

In [4] daisy cubes were defined as follows. Let $\leq$ be a partial order on $B^{n}$ defined in the following way: $u_{1}u_{2}\ldots u_{n}\leq v_{1}v_{2}\ldots v_{n}$ if $u_{i}\leq v_{i}$ is true for all $i\in\{1,2,\ldots,n\}$. For $X\subseteq B^{n}$ the graph $Q_{n}(X)$ is defined as the subgraph of $Q_{n}$ where

[TABLE]

The graph $Q_{n}(X)$ is called a daisy cube (generated by $X$).

As noted in [4], if $x,y\in X$ are such that $y\leq x$, then $Q_{n}(X)=Q_{n}(X\setminus\{y\})$.

Proposition 1.2**.**

[4]** If $X\subseteq B^{n}$, then $Q_{n}(X)$ is a partial cube.

Several characterizations of partial cubes and median graphs are known. Before we state some, we will need the notion of expansion.

Suppose $V(G)=V_{1}\cup V_{2}$, where $V_{1}\cap V_{2}\not=\emptyset$, both $\langle V_{1}\rangle$ and $\langle V_{2}\rangle$ are isometric subgraphs of $G$, and there exists no edge of $G$ with one endpoint in $V_{1}\setminus V_{2}$ and the other in $V_{2}\setminus V_{1}$. An expansion of $G$ with respect to $V_{1}$ and $V_{2}$ is a graph $G^{\prime}$ obtained from $G$ by the following steps:

1. (i)

Replace each $v\in V_{1}\cap V_{2}$ with vertices $v_{1},v_{2}$, and add the edge $v_{1}v_{2}$. 2. (ii)

Add edges between $v_{1}$ and all neighbours of $v$ in $V_{1}\setminus V_{2}$, also add edges between $v_{2}$ and all neighbours of $v$ in $V_{2}\setminus V_{1}$. 3. (iii)

Insert the edges $v_{1}u_{1}$ and $v_{2}u_{2}$ if $u,v\in V_{1}\cap V_{2}$ are adjacent in $G$.

An expansion is called a connected expansion if $\langle V_{1}\cap V_{2}\rangle$ is a connected subgraph of $G$. An expansion is called a convex expansion if $\langle V_{1}\cap V_{2}\rangle$ is a convex subgraph of $G$. An expansion is peripheral if $V_{1}=V(G)$. We will denote peripheral expansions by $\mathop{\rm pe}\nolimits(G;V_{2})$ and call them peripheral expansions of $G$ with respect to $V_{2}$.

The inverse operation of an expansion is called a contraction. Note, a contraction of a partial cube is obtained by contracting the edges of a given $\Theta$-class.

The following characterization of partial cubes is well known.

Theorem 1.3**.**

[1]** A graph $G$ is a partial cube if and only if $G$ can be obtained from the one-vertex graph by a sequence of expansions.

Median graphs are also characterized in a manner similar to Theorem 1.3. This characterization is known as the Mulder’s Convex Expansion Theorem:

Theorem 1.4**.**

[2, Theorem 12.8]** A graph $G$ is a median graph if and only if it can be obtained from the one-vertex graph by a sequence of convex expansions.

In the next section we will characterize daisy cubes in the language of expansions, similar to Theorem 1.3 and Theorem 1.4. For this, we will need the following result.

Proposition 1.5**.**

[4]** If $G=Q_{n}(X)$ is a daisy cube, then a contraction of $G$ is a daisy cube.

2 Characterization

Let $G$ be a graph isomorphic to a daisy cube $Q_{h}(X)$. There can be more than one isometric embedding of $G$ into the hypercube $Q_{h}$ (each such embedding assigns corresponding labelling to vertices of $G$). Let $X_{G}\subseteq B^{h}$ be the set of labels of the vertices of $G$ assigned by an isometric embedding of $G$ into $Q_{h}$. We say that $G$ has a proper labelling if $G$ is isomorphic to $Q_{h}(X_{G})$. Otherwise the labelling is improper.

In Figure 1(a) the graph $G$ is isomorphic to the daisy cube $Q_{3}(\{011,100\})$. In Figure 1(b) we see two other embeddings of the graph $G$ into the hypercube $Q_{3}$. The embedding shown with thick dashed lines is defined by: $u_{1}\mapsto 000$, $u_{2}\mapsto 001$, $u_{3}\mapsto 101$, $u_{4}\mapsto 100$ and $u_{5}\mapsto 010$. This embedding assigns a proper labelling to the vertices of $G$. The embedding shown with thick solid lines is defined by: $u_{1}\mapsto 110$, $u_{2}\mapsto 010$, $u_{3}\mapsto 011$, $u_{4}\mapsto 111$ and $u_{5}\mapsto 100$, corresponds to an improper labelling (e.g. $101\leq 111$ and $101$ is not a label of any vertex in $G$).

Since the embedding of the daisy cube $Q_{h}(X)$ induced by the definition of a daisy cube gives a proper labelling of its vertices, from here on we can always assume that labellings considered in graphs isomorphic to a daisy cube are proper labellings. Later on, we will also provide an algorithms which for a given unlabelled graph $G$ isomorphic to some daisy cube assigns a proper labelling to the vertices of $G$.

We now continue with results leading to a characterization of daisy cubes.

Proposition 2.1**.**

Every $\Theta$-class of a daisy cube $G$ is peripheral.

Proof.

Let $G$ be a daisy cube. Consider an arbitrary edge $e=ab$ of $G$. Its endpoints differ in exactly one position, say position $i$. We choose the notation such that the endpoint of $e$ with 0 at position $i$ is denoted by $a$, and the other endpoint is denoted by $b$. Let $E=\{f\ |\ f\in E(G)\text{ and }e\Theta f\}$. We claim that $E$ is peripheral with $U_{ba}=W_{ba}$.

Since $G$ is a partial cube, $W_{ab}$ and $W_{ba}$ partition $V(G)$ into the sets that contain all vertices of $G$ with 0 at position $i$ and all vertices of $G$ with 1 at position $i$, respectively. Let $x:=b_{1}b_{2}\ldots b_{i-1}1b_{i+1}\ldots b_{n}\in W_{ba}$ be arbitrarily chosen. Let $y:=b_{1}b_{2}\ldots b_{i-1}0b_{i+1}\ldots b_{n}$. Clearly $y\in V(G)$, since $y\leq x$, and $y\in W_{ab}$, since it has a 0 at position $i$. The vertices $x$ and $y$ differ in exactly one bit (the one at position $i$), therefore $xy\in E(G)$. This means that every vertex of $W_{ba}$ is adjacent to a vertex in $W_{ab}$. From definition of $U_{ba}$ it follows that $W_{ba}=U_{ba}$. ∎

Proposition 2.2**.**

Every $\Theta$-class of a daisy cube $G$ contains an edge with the vertex $0^{n}$ as an endpoint.

Proof.

Let $ab\in E(G)$ be arbitrary, with notation chosen such that the endpoint $a$ has 0 at position $i$ and the endpoint $b$ has 1 at position $i$. Consider the vertices $u=0^{n}$ and $v=0^{i-1}10^{n-i}$. Since $u\leq b$ and $v\leq b$, both $u$ and $v$ are vertices of $G$. It follows that $uv\in E(G)$. It is trivial to verify that $ab\Theta uv$. ∎

The following Corollary immediately follows from Proposition 1.2 and Proposition 2.2.

Corollary 2.3**.**

If $G$ is a daisy cube, then the vertex $0^{n}$ is a vertex of degree $\Delta(G)$.

Proposition 2.4**.**

For any edge $ab$ of a daisy cube $G=Q_{n}(X)$ the graphs $\langle W_{ab}\rangle$ and $\langle W_{ba}\rangle$ are daisy cubes.

Proof.

Let $e=ab$ be an arbitrary edge of $G$, with notation chosen such that the endpoint $a$ has 0 at position $i$ and the endpoint $b$ has 1 at position $i$. Then the set $W_{ab}$ consists of all vertices of $G$ with 0 at position $i$ and $W_{ba}$ consists of all vertices of $G$ with a 1 at position $i$.

Let $E=\{f\ |\ f\in E(G)\text{ and }e\Theta f\}$. Using Proposition 2.1 it follows that $\langle W_{ab}\rangle$ is obtained by contracting the edges of the $\Theta$-class $E$. From Proposition 1.5 it follows that $\langle W_{ab}\rangle$ is a daisy cube.

To prove that $\langle W_{ba}\rangle$ is a daisy cube consider the following. Let $W^{\prime}_{ba}=\{x_{1}\ldots x_{i-1}x_{i+1}\ldots x_{n}\ |\ x_{1}\ldots x_{n}\in W_{ba}\}$. Since all vertices of $W_{ba}$ have 1 at position $i$, it is easy to see that the function $r$ that removes the bit at position $i$ from a binary string of length $n$ is a bijection between $W_{ba}$ and $W^{\prime}_{ba}$.

Let $u=u_{1}u_{2}\ldots u_{n-1}\in W^{\prime}_{ba}$ be arbitrarily chosen. We claim that any binary string $v$ of length $n-1$, such that $v\leq u$, belongs to $W^{\prime}_{ba}$. Towards contradiction suppose there is a binary string $w=w_{1}w_{2}\ldots w_{n-1}$ with $w\leq u$ and $w\not\in W^{\prime}_{ba}$. Since $u_{1}\ldots u_{i-1}1u_{i}\ldots u_{n-1}\in W_{ba}$ and $w_{1}\ldots w_{i-1}1w_{i}\ldots w_{n-1}\leq u_{1}\ldots u_{i-1}1u_{i}\ldots u_{n-1}$ it follows that $w_{1}\ldots w_{i-1}1w_{i}\ldots w_{n-1}\in W_{ba}$. But then $w\in W^{\prime}_{ba}$, a contradiction. Therefore for any $x\in W^{\prime}_{ba}$ the interval $I_{Q_{n-1}}(x,0^{n-1})\subseteq W^{\prime}_{ba}$. It follows that $\langle W^{\prime}_{ba}\rangle_{Q_{n-1}}$ is a daisy cube. The assertion follows from the fact that $r$ defines an isomorphism between $\langle W_{ba}\rangle_{Q_{n}}$ and $\langle W^{\prime}_{ba}\rangle_{Q_{n-1}}$. ∎

Let $G$ be a daisy cube. An induced subgraph $H$ of the graph $G$ is called a $\leq$-subgraph if $V(H)=\{u\in V(G)\ |\ u\leq v\text{ for some }v\in V(H)\}$.

The next proposition follows immediately from the definition of a $\leq$-subgraph.

Proposition 2.5**.**

Let $G$ be a daisy cube. If $H$ is a $\leq$-subgraph of $G$, then $H$ is isomorphic to a daisy cube.

The converse of Proposition 2.5 may not be necessarily true as the example in Figure 2 shows. In Figure 2(a) we see the daisy cube $Q_{4}(X)$, where $X=\{0011,0110,1100,1001\}$. The subgraph $H_{1}$ induced on the vertex set $\{0010,0000,1000\}$ is a $\leq$-subgraph, and is isomorphic to $Q_{2}(\{01,10\})$. On the other hand, the subgraph $H_{2}$ is also isomorphic to $Q_{2}(\{01,10\})$, however it is not a $\leq$-subgraph of $Q_{4}(X)$. Figure 2(b) shows the daisy cube $Q_{4}(Y)$, where $Y=\{0011,1000,0100\}$. The subgraph $H$ of $Q_{4}(Y)$ is isomorphic to the daisy cube $Q_{2}(\{01,10\})$ and it is not a $\leq$-subgraph of $Q_{4}(Y)$.

Corollary 2.6**.**

Let $G$ be a daisy cube and $ab\in E(G)$ arbitrary. If the notation is chosen such that $a=a_{1}\ldots a_{i-1}0a_{i+1}\ldots a_{n}$ and $b=a_{1}\ldots a_{i-1}1a_{i+1}\ldots a_{n}$, then $\langle U_{ab}\rangle$ is a $\leq$-subgraph of $G$.

Proof.

Towards contradiction suppose $\langle U_{ab}\rangle$ is not a $\leq$-subgraph. Proposition 2.1 says that $W_{ba}=U_{ba}$. Then for some vertex of $x=x_{1}\ldots x_{n}\in U_{ab}$ there is a vertex $y=y_{1}\ldots y_{n}\in B^{n}$ such that $y\leq x$ and $y\not\in U_{ab}$. Note that $x_{i}=0$ and since $y\leq x$, also $y_{i}=0$. Since all vertices in relation $\leq$ with $x$ are in $G$ (by definition of a daisy cube), then $y\in W_{ab}\setminus U_{ab}$. Let $x^{\prime}=x_{1}\ldots x_{i-1}1x_{i+1}\ldots x_{n}$ and $y^{\prime}=y_{1}\ldots y_{i-1}1y_{i+1}\ldots y_{n}$. Since $x\in U_{ab}$, and the set $U_{ba}$ contains only vertices with 1 at position $i$, and $x$ and $x^{\prime}$ differ in exactly one position, it follows that $x^{\prime}\in U_{ba}$. The fact $y\leq x$ implies that $y^{\prime}\leq x^{\prime}$, moreover $y^{\prime}\in U_{ba}$, since $y_{i}=1$. The vertices $y$ and $y^{\prime}$ differ in exactly one position, which means that $yy^{\prime}\in E(G)$, further implying $y\in U_{ab}$, a contradiction. ∎

Let $H$ be a $\leq$-subgraph of a daisy cube $G$. Then the peripheral expansion $\mathop{\rm pe}\nolimits(G;V(H))$ is called the $\leq$-expansion of $G$ with respect to $H$.

Proposition 2.7**.**

Let $G=Q_{h}(X)$ be a daisy cube and $H$ a $\leq$-subgraph of $G$. The $\leq$-expansion of $G$ with respect to $H$ is a daisy cube.

Proof.

Let $G^{\prime}=\mathop{\rm pe}\nolimits(G;V(H))$. Then $G^{\prime}$ consists of a disjoint union of a copy of $G$ and a copy $H$ with an edge between each vertex of $H$ and the corresponding vertex in the copy of $H$. Let the labels of the vertices of $G^{\prime}$ be defined as follows. Prepend a 0 to the label of each vertex in $G^{\prime}$ corresponding to the copy of $G$ and a 1 to each vertex of $G^{\prime}$ corresponding to the copy of $H$. So the labels of the vertices of $G^{\prime}$ are binary strings of length $h+1$. We prove the assertion in two steps: showing first, that the vertex set of $G^{\prime}$ is the vertex set of a daisy cube, and concluding the proof by showing that two vertices are adjacent if and only if they differ in exactly one bit.

First, we will show that for every $v\in V(G^{\prime})$ all the vertices in relation $\leq$ with $v$ are also in $V(G^{\prime})$. Let $v\in V(G^{\prime})$ be arbitrary. If $v=0v_{1}\ldots v_{h}$ ($v$ is in the copy of $G$ in $G^{\prime}$) then any binary string $u$ of length $h+1$, such that $u\leq v$, is also of the form $0u_{1}\ldots u_{h}$. Since all binary strings $w_{1}\ldots w_{h}$ of length $h$ such that $w_{1}\ldots w_{h}\leq v_{1}\ldots v_{h}$ are in $G$, the assertion for this case follows. Now, let $v=1v_{1}\ldots v_{h}$ ($v$ is in the copy of $H$ in $G^{\prime}$). Since $H$ is a $\leq$-subgraph, following the same line of thought as in the previous case, we can show that all binary strings $u=1u_{1}\ldots u_{h}$ such that $u\leq v$, are also in $V(G^{\prime})$. To see, that also all binary strings $u=0u_{1}\ldots u_{h}$ such that $u\leq v$ are in $V(G^{\prime})$, remember that there is and edge between $v$ (which is in the copy of $H$) and a vertex $w$ in the copy of $G$. Since $v=1v_{1}\ldots v_{h}$ and $w=0w_{1}\ldots w_{h}$ and $vw\in E(G^{\prime})$ it follows that for every $i\in\{1,\ldots,h\}$ the values $v_{i}$ and $w_{i}$ are equal. From the first part, all vertices in relation $\leq$ with $w$ are in $V(G^{\prime})$ and these are exactly the vertices $u=0u_{1}\ldots u_{h}$ such that $u\leq v$.

Second, we show that two vertices are adjacent if they differ in exactly one position. Take two arbitrary adjacent vertices of $G^{\prime}$, say $v=v_{0}v_{1}\ldots v_{h}$ and $u=u_{0}u_{1}\ldots u_{h}$. If both $v_{0}$ and $u_{0}$ are 0 (or 1), then $u$ and $v$ correspond to two adjacent vertices in the copy of $G$ (or $H$), which is an induced subgraph of $Q_{h}$ (is a $\leq$-subgraph), and the assertion follows. If $0=v_{0}\not=u_{0}=1$, then there is an edge between the two vertices if and only if $v_{1}\ldots v_{h}$ is a vertex of $H$ in $G$ and $u_{1}\ldots u_{h}$ is the corresponding vertex in the copy of $H$, meaning that for every $i\in\{1,\ldots,h\}$ it holds that $v_{i}=u_{i}$. This completes the proof. ∎

Using Proposition 2.6 and Proposition 2.7 we immediately obtain the following characterization of daisy cubes.

Theorem 2.8**.**

A connected graph $G$ is a daisy cube if and only if it can be obtained from the one-vertex graph by a sequence of $\leq$-expansions.

From this characterization and Theorem 1.4 we can also characterize all median daisy cubes as follows.

Corollary 2.9**.**

A daisy cube is a median graph if and only if it can be obtained from the one-vertex graph by a sequence of convex $\leq$-expansions.

3 Finding proper labellings

In this section we present an algorithm which for a given unlabelled graph $G$ isomorphic to a daisy cube (the embedding into the corresponding hypercube is not given) assigns a proper labelling to vertices of $G$.

Proposition 3.1**.**

Let $G=Q_{n}(X)$ be a daisy cube and $e=ab\in E(G)$. If the notation of the endpoints of $e$ can be chosen such that $|W_{ab}|>|W_{ba}|$, then $0^{n}\in W_{ab}$.

Proof.

By Corollary 2.6, the notation can be chosen such that $a$ has a 0 at position $i$ and $b$ has a 1 at position $i$ and $U_{ab}$ induces a $\leq$-subgraph. For any vertex $u\in V(G)$ it holds that $0^{n}\leq u$ and therefore $0^{n}\in U_{ab}\subseteq W_{ab}$. ∎

Proposition 3.2**.**

Let $G=Q_{n}(X)$ be a daisy cube and $e=ab\in E(G)$. If $|W_{ab}|=|W_{ba}|$, then there exists a proper labelling of $G$ such that $0^{n}\in W_{ab}$ and a proper labelling of $G$ such that $0^{n}\in W_{ba}$.

Proof.

The fact that $|W_{ab}|=|W_{ba}|$ and that every $\Theta$-class in $G$ is peripheral (Proposition 2.1) implies that $W_{ab}=U_{ab}$ and $W_{ba}=U_{ba}$, moreover the $\Theta$-equivalence class of $ab$ induces a matching (and also an isomorphism of corresponding induced subgraphs) between $U_{ab}$ and $U_{ba}$. Therefore $G$ is isomorphic to $\langle U_{ab}\rangle_{G}\Box K_{2}$. By Corollary 2.6 at least one of $U_{ab}$ and $U_{ba}$ induces a $\leq$-subgraph. W.l.o.g., assume it is $U_{ab}$, therefore $0^{n}\in U_{ab}$. Let $u\in U_{ba}$ be the vertex adjacent to $0^{n}$. It follows that $u$ has exactly one 1, say at position $i$. Moreover, all vertices in $U_{ab}$ have 0 at position $i$ and all vertices of $U_{ba}$ have 1 at position $i$. Changing values at position $i$ to 1 for all vertices of $U_{ab}$ and to 0 for all vertices of $U_{ba}$ we obtain another labelling of the vertices of $G$. It is easy to verify that such labelling is a proper labelling and that in this case $0^{n}$ belongs to $U_{ba}$ and therefore to $W_{ba}$. ∎

From Proposition 3.1 and Proposition 3.2 we obtain the following corollary.

Corollary 3.3**.**

Let $G=Q_{n}(X)$ be a daisy cube and $e=ab\in E(G)$. If the notation can be chosen such that $|W_{ab}|\geq|W_{ba}|$, then there exists a proper labelling of $G$ such that $0^{n}\in W_{ab}$.

These results enable us to find a proper labelling for any graph isomorphic to a daisy cube as presented in Algorithm 1.

Theorem 3.4**.**

Algorithm 1 assigns a proper labelling to vertices of an unlabelled graph isomorphic to a daisy cube in $O(mn)$ time, where $n$ is the number of vertices and $m$ the number of edges of $G$.

Proof.

The correctness of the assigned labels follows from Proposition 3.1 and Corollary 3.3.

Now consider the time complexity. Step 1 can be done in $O(mn)$ time [2, Theorem 18.6]. For one iteration of the loop at step 2 the following holds. The edge $ab$ can be chosen in constant time. The sets $W_{ab}$ and $W_{ba}$ can be computed in $O(m)$ time (e.g. by using BFS from each endpoint of $ab$). Both loops (step 12 and step 15) run together in $O(n)$ time. Since the number of $\Theta$-classes (the number of times the for loop at step 2 repeats) is bounded by $n$, the assertion follows. ∎

4 Generalization of daisy cubes

In this section we give a generalization of the concept of daisy cubes. All graphs considered are connected.

Definition 4.1**.**

Let $G$ be a rooted graph with the root $r$. Let $u$ and $v$ be two vertices of $G$. We say that $u\leq_{G,r}v$, if $u$ is on some shortest $v,r$-path.

Proposition 4.2**.**

Let $G$ be a rooted graph with the root $r$. The relation $\leq_{G,r}$ is a partial order on $V(G)$.

Proof.

Since every vertex $v\in V(G)$ is on every shortest path from $v$ to $r$, it follows that $\leq_{G,r}$ is reflexive.

Assume that for any two vertices $u$ and $v$ from $V(G)$ it holds that $u\leq_{G,r}v$ and $v\leq_{G,r}u$. This means that $u$ is on some shortest $v,r$-path and that $v$ is on some shortest $u,r$-path. Therefore, $d(v,r)=d(v,u)+d(u,r)$ and $d(u,r)=d(u,v)+d(v,r)$. This implies that $d(u,v)=0$, meaning that $u=v$. Therefore $\leq_{G,r}$ is antisymmetric.

Let $u,v$ and $w$ be arbitrary vertices of $G$. Also, let $u\leq_{G,r}v$ and $v\leq_{G,r}w$. This means $d(v,r)=d(v,u)+d(u,r)$ and that $d(w,r)=d(w,v)+d(v,r)$. This gives that $d(w,r)=d(w,v)+d(v,u)+d(u,r)$ which implies that $u$ is on some shortest $w,r$-path and therefore $u\leq_{G,r}w$. This proves the transitivity property and concludes the proof. ∎

Definition 4.3**.**

Let $G$ be a rooted graph with the root $r$. For $X\subseteq V(G)$ the daisy graph $G_{r}(X)$ of the graph $G$ with respect to $r$ (generated by $X$) is the subgraph of $G$ where

[TABLE]

From Definition 4.3 it immediately follows that if $u$ is a vertex of the daisy graph $G_{r}(X)$ of the graph $G$ with respect to $r$, then $I_{G}(u,r)\subseteq V(G_{r}(X))$. This fact also immediately gives the following result.

Proposition 4.4**.**

Let $G$ be a rooted graph with the root vertex $r$. If $H$ is a convex subgraph of $G$, such that $r\in V(H)$, then $H$ is a daisy graph of $G$ with respect to $r$.

By definition of relation $\leq$ on the vertices of the hypercube $Q_{h}$ (binary strings of length $h$), one immediately obtains that for two such vertices, say $u$ and $v$, the following is true: $u\leq v$ if and only if $u$ is on some shortest path between $v$ and $0^{h}$, which is definition of $\leq_{Q_{h},0^{h}}$. Therefore daisy cubes are a special case of daisy graphs, specifically as the following proposition says.

Proposition 4.5**.**

If $G=Q_{h}(X)$ is a daisy cube, then $G$ is a daisy graph of $Q_{h}$ with respect to $0^{h}$.

As stated in Proposition 1.2, daisy cubes are isometric subgraphs of hypercubes. We now give a sufficient condition for when a daisy graph of a rooted graph $G$ with respect to the root $r$ is isometric.

Proposition 4.6**.**

Let $G$ be a rooted graph with the root $r$. If for any two vertices of $G$, say $u$ and $v$, it holds that there exists a median of $u$, $v$ and $r$, then every daisy graph of $G$ with respect to $r$ is isometric in $G$.

Proof.

Let $H$ be an arbitrary daisy graph of $G$ with respect to $r$. Also, let $u$ and $v$ be two arbitrary vertices of $H$, and let $w$ be a median of $u$, $v$ and $r$. Since $I_{G}(u,r)\subseteq V(H)$, $I_{G}(v,r)\subseteq V(H)$ and $H$ is an induced subgraph of $G$, then any shortest $u,r$-path and any shortest $v,r$-path is completely contained in $H$. Also, $w$ is on some shortest $u,r$-path, as well as on some shortest $v,r$-path. This implies that $d_{G}(u,w)=d_{H}(u,w)$ and $d_{G}(v,w)=d_{H}(v,w)$. We know that $w$ is also on some shortest $u,v$-path in $G$, implying $d_{G}(u,v)=d_{G}(u,w)+d_{G}(w,v)$. Also, $d_{H}(u,v)\leq d_{H}(u,w)+d_{H}(w,v)=d_{G}(u,w)+d_{G}(w,v)=d_{G}(u,v)$. Since $H$ is a subgraph of $G$ we also know that $d_{H}(u,v)\geq d_{G}(u,v)$, thus $d_{H}(u,v)=d_{G}(u,v)$. Since this holds for two arbitrary vertices $u$ and $v$, it follows that $H$ is an isometric subgraph of $G$. ∎

The converse of Proposition 4.6 may not necessarily be true. In Figure 3 we see a rooted graph $G$ with the root $r$ and the daisy graph $H$ of the graph $G$ with respect to $r$ generated by $\{u,v\}$. The edges of $H$ are depicted with thick lines. $H$ is an isometric subgraph of $G$, however, no median exists for the vertices $u$, $v$ and $r$.

Corollary 4.7**.**

Let $G$ be a rooted median graph with the root $r$. Every daisy graph of $G$ with respect to $r$ is isometric in $G$.

Proof.

Since in median graphs every triple of vertices admits a (unique) median, then for two arbitrary vertices of $G$ and the root $r$ there also exists a median. By Proposition 4.6 the assertion follows. ∎

5 Conclusion

In this paper we answered an open problem by Klavžar and Mollard proposed in [4]. Namely, we provide a characterization of daisy cubes in terms of an expansion procedure. We also presented some interesting properties of daisy cubes and provide an $O(mn)$ time algorithm for finding a proper embedding of a daisy cube into the corresponding hypercube. Also, we proposed a generalization of daisy cubes and provided some exciting results concerning these graphs.

Moreover, the following further investigations might be interesting.

Problem 5.1**.**

Find a non-constructive characterization of daisy cubes.

Problem 5.2**.**

Is there a faster way of finding the vertex $0^{h}$ of a daisy cube $Q_{h}(X)$ than the one provided in Algorithm 1?

A positive answer to Problem 5.2 would give a linear time algorithm for finding a proper labelling of a graph isomorphic to a daisy cube.

Problem 5.3**.**

Provide a recognition algorithm for daisy cubes.

Proposition 4.6 provides a condition, such that if a rooted graph $G$ with the root $r$ satisfies this condition, then every daisy graph of the graph $G$ with respect to $r$ is isometric.

Problem 5.4**.**

Characterize rooted graphs $G$ with the root $r$ for which every daisy graph of a rooted graph $G$ with respect to $r$ is isometric.

Acknowledgments

This work was supported by the Slovenian Research Agency under the grants P1-0297 and J1-9109.