The Complexity of All $(g,f)$-Factor Problem

TL;DR

This paper proves that determining whether a graph has all $(g,f)$-factors is NP-hard, answering a longstanding open question and showing the problem's computational complexity.

Contribution

It establishes the NP-hardness of testing for all $(g,f)$-factors in graphs, providing a negative answer to Niessen's 1998 open problem.

Findings

Proves the problem is NP-hard.

Answers Niessen's open question negatively.

Highlights computational difficulty of all $(g,f)$-factor problem.

Abstract

Let $G$ be a graph with vertex set $V$ and let $g, f : V\rightarrow \mathbb{Z}^+$ be two functions such that $g\le f$. We say that $G$ has all $(g, f )$-factors if $G$ has an $h$-factor for every $h: V\rightarrow \mathbb{Z}^+$ such that $g(v)\le h(v)\le f (v)$ for every $v\in V$ and $\sum_{v\in{V}}h(v)\equiv 0\pmod 2$. Two decades ago, Niessen derived from Tutte's $f$-factor theorem a similar characterization for the property of graphs having all $(g, f )$-factors and asked whether there is a polynomial time algorithm for testing whether a graph $G$ has all $(g, f )$-factors (A characterization of graphs having all $(g, f )$-Factors, \emph{J. Combin. Theory, Ser. B}, \textbf{72} (1998), 152--156). In this paper, we show that it is NP-hard to determine whether a graph $G$ has all $(g,f)$-factors, which gives a negative answer to the question of Niessen.

Equations

$$\omega (G-S)\le |S|+1.$$

$$H^{-1}(1):=\{v\in V(G):H(v)=\{1\}\}.$$

$$g(D)-f(S)+d_{G-D}(S)-q_{G}(D,S,g,f)\geq\left\{\begin{array}[]{ll}-1,&\hbox{$g\neq f$;}\\ 0,&\hbox{$g=f$.}\end{array}\right.$$

$$h_{H}(u)=\left\{\begin{array}[]{ll}2,&\hbox{if $H(u)=\{0,2\}$ and $u\in V(G)$;}\\ 1,&\hbox{otherwise.}\end{array}\right.$$

$$\omega (G^x-S)\le \omega (G-S)+|\{x^{\prime }\}|\le |S|+1.$$

$$\omega (G^x-S)=\omega (G-S)+1\ge |S|+2,$$

$$h_{H}(x)=d_{F^{\prime}}(x)=\left\{\begin{array}[]{ll}2,&\hbox{if $H(x)=\{0,2\}$ and $x\in V(G)$;}\\ 1,&\hbox{otherwise.}\end{array}\right.$$

$$H(u)=\left\{\begin{array}[]{ll}\{1\},&\hbox{$h_{H}(u)=1$ and $u\in V(G)$;}\\ \{0,2\},&\hbox{$h_{H}(u)=2$ and $u\in V(G)$.}\end{array}\right.$$

$$f(u)=\left\{\begin{array}[]{ll}2,&\hbox{if $u\in V(G)$;}\\ 1,&\hbox{else $u\in V(G_{L})-V(G)$.}\end{array}\right.$$

$$\mathcal{H}=\{H\ |\ \mbox{$H:V(G)\rightarrow\{\{1\},\{0,2\}\}$ such that $|H^{-1}(1)|$ is even}\}.$$

$$\mathcal{F}=\{h\ |\ h:V(G_{L})\rightarrow\{1,2\}\\ \mbox{ such that $|h^{-1}(1)|$ is even and $h(v)=1$ $\forall$ $v\in X\cup Y$}\}.$$

$$H(v)=\left\{\begin{array}[]{ll}\{1\},&\hbox{if $h(v)=1$ and $v\in V(G)$;}\\ \{0,2\},&\hbox{else $h(v)=2$ and $v\in V(G)$.}\end{array}\right.$$

$$f(u)=\left\{\begin{array}[]{ll}2,&\hbox{if $u\in V(G)$;}\\ 1,&\hbox{else $u\in V(G_{L})-V(G)$.}\end{array}\right.$$

Full text

1 Introduction

We consider finite simple graphs, i.e., undirected graph without loops or multiple edges. Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$.

Let $\omega(G)$ denote the number of components of a graph $G$. A graph $G$ is $t$-tough if $|S|\leq t\omega(G-S)$ for every subset $S$ of the vertex set $V(G)$ with $\omega(G-S)>1$. The toughness of $G$, denoted $t(G)$, is the maximum value of $t$ for which $G$ is $t$-tough (taking $t(K_{n})=\infty$ for all $n\leq 1$). We say that a graph $G$ is almost $1$-tough if for any $S\subseteq V(G)$,

[TABLE]

Let $H:V(G)\rightarrow 2^{\mathbb{Z}^{+}}$ be a set function. An $H$-factor $F$ is a spanning subgraph such that $d_{F}(v)\in H(v)$ for every $v\in V(G)$. Given two integer-valued functions $g,f:V(G)\rightarrow\mathbb{Z}^{+}$ such that $g(v)\leq f(v)$ for all $v\in V(G)$, an $H$-factor is also called $(g,f)$-factor if $H(v)=[g(v),f(v)]$ for all $v\in V(G)$. In particular, if $g=f$, then an $(f,f)$-factor is also called an $f$-factor. We say that $G$ has all $(g,f)$-factors if for any integer-valued function defined on $V(G)$ such that $h(V(G))\equiv 0\pmod{2}$ and $g(v)\leq h(v)\leq f(v)$ for all $v\in V(G)$, $G$ contains an $h$-factor. We write

[TABLE]

For simplicity, we write $g^{-1}(1):=H^{-1}(1)$ if $H(v)=\{g(v)\}$ for all $v\in V(G)$.

Niessen [7] derived from Tutte’s $f$-factor theorem a similar characterization for the property of graphs having all $(g,f)$-factors.

Theorem 1** (Niessen [7])**

$G$* has all $(g,f)$-factors if and only if*

[TABLE]

for all disjoint sets $D,S\subseteq V$, where $q_{G}(D,S,g,f)$ denotes the number of components $C$ of $G-(D\cup S)$ such that there exists a vertex $v\in V(C)$ with $g(v)<f(v)$ or $e_{G}(V(C),S)+f(V(C))\equiv 1\pmod{2}$.

It is well-known that there exists a polynomial time algorithm to determine whether a graph has a $(g,f)$-factor. An open problem was naturally proposed in [7]:

Problem 2

Is there a polynomial time algorithm for testing whether a graph $G$ has all $(g,f)$-factors?

In this note, we obtain the following theorem, which gives a negative answer to Problem 2, unless P=NP.

Theorem 3

It is NP-hard to determine whether a graph $G$ has all $(g,f)$-factors.

The main ingredient in the proof of Theorem 3 is the following two results.

Theorem 4** (Bauer et. al., [3])**

It is NP-hard to recognize 1-tough cubic graphs.

Theorem 5** (Kano and Lu, [6])**

Let $G$ be a connected graph. A graph $G$ has an $H$-factor for every $H:V(G)\rightarrow\{\{1\},\{0,2\}\}$ with $|H^{-1}(1)|$ even if and only if $G$ is almost 1-tough.

Theorem 5 gives a polynomial time reduction from the toughness of graphs to the degree constrained factors.

2 The Proof of Theorem 3

To prove Theorem 3, we need the following notations.

Definition 6

Let $G$ be a graph with $V(G)=\{v_{1},\ldots,v_{n}\}$. Write $X=\{x_{1},\ldots,x_{n}\}$, $Y=\{y_{1},\ldots,y_{n}\}$.

• (i)

For any vertex $x$ of $G$, let $G^{x}$ denote the graph obtained from $G$ by adding a new vertex $x^{\prime}$ together with a new edge $xx^{\prime}$, that is, $G^{x}=G+xx^{\prime}$.

• (ii)

Let $G_{L}$ be a graph with vertex set $V(G_{L})$ and edge set $E(G_{L})$, where $V(G_{L})=V(G)\cup X\cup Y$ and $E(G_{L})=E(G)\cup\{x_{i}y_{i},y_{i}v_{i},v_{i}x_{i}\ |\ i\in[n]\}$;

• (iii)

For any $H:V(G)\rightarrow\{\{1\},\{0,2\}\}$, we define $h_{H}:V(G_{L})\rightarrow\{1,2\}$ such that

[TABLE]

Now we show the following result.

Lemma 7

A graph $G$ is 1-tough if and only if for any $x\in V(G)$, $G^{x}$ is almost 1-tough.

Proof. Necessity. Suppose that $G$ is 1-tough. Let $x\in V(G)$. For any $S\subseteq V(G^{x})$,

[TABLE]

Sufficiency. By contradiction, suppose that $G$ is not 1-tough. Then there exists $\emptyset\neq S\subseteq V(G)$ such that $\omega(G-S)\geq|S|+1$. Let $x\in S$. One can see that

[TABLE]

which contradicts the fact that $G$ is 1-tough.  $\Box$

From Theorem 4 and Lemma 7, one may see that

Lemma 8

It is NP-hard to recognize almost 1-tough cubic graph.

Lemma 9

Let $G$ be a connected cubic graph and let $H:V(G)\rightarrow\{\{1\},\{0,2\}\}$. Then $G$ contains an $H$-factor if and only if $G_{L}$ contains an $h_{H}$-factor.

Proof. Necessity. Suppose that $G$ contains an $H$-factor $F$. Define M_{1}=\{x_{i}y_{i}\ |\ d_{F}(v_{i})\in\{1,2\}\ \mbox{for i\in[n]}\} and M_{2}=\{x_{i}v_{i},y_{i}v_{i}\ |\ d_{F}(v_{i})=0\ \mbox{for i\in[n]}\}. Let $F^{\prime}$ be a spanning subgraph of $G_{L}$ with edge set $E(F)\cup M_{1}\cup M_{2}$. From the definition of function $h_{H}$, one can see that

[TABLE]

So $F^{\prime}$ is an $h_{H}$-factor of $G_{L}$.

Sufficiency. Suppose that $F^{\prime}$ be an $h_{H}$-factor of $G_{L}$. Let $F$ be a spanning subgraph of $G$ with edge set $E(F^{\prime})\cap E(G)$. Now we show that $F$ is an $H$-factor of $G$. From Definition 6 (iii), one can see that

[TABLE]

Consider $h_{H}(v_{i})=d_{F^{\prime}}(v_{i})=1$. Since $d_{F^{\prime}}(x_{i})=d_{F^{\prime}}(y_{i})=1$, then we have $x_{i}v_{i},y_{i}v_{i}\notin E(F^{\prime})$. So we have $d_{F}(x_{i})=1\in H(v_{i})$. Next we may assume that $d_{F^{\prime}}(v_{i})=2$. Then either $\{x_{i}v_{i},y_{i}v_{i}\}\subseteq E(F^{\prime})$ or $\{x_{i}v_{i},y_{i}v_{i}\}\cap E(F^{\prime})=\emptyset$. In the former case, we have $d_{F}(v_{i})=0$, and in the latter case we have $d_{F}(v_{i})=2$. So in both cases, one can see that $d_{F}(v_{i})\in\{0,2\}$ and $F$ is an $H$-factor of $G$. This completes the proof.  $\Box$

Lemma 10

Let $G$ be a connected cubic graph. Then $G$ contains an $H$-factor for any $H:V(G)\rightarrow\{\{1\},\{0,2\}\}$ with $|H^{-1}(1)|$ even if and only if $G_{L}$ contains all $(g,f)$-factors, where $g\equiv 1$ and

[TABLE]

Proof. Define

[TABLE]

and

[TABLE]

Let $J:\mathcal{H}\rightarrow\mathcal{F}$ such that $J(H)=h_{H}$ for all $H\in\mathcal{H}$. From the definition of $h_{H}$, one can see that $J$ is well-defined. By Lemma 9, it is suffices for us to show that $J$ is a bijection. Firstly, we show that $J$ is injective. For any $H,H^{\prime}\in\mathcal{H}$ such that $H\neq H^{\prime}$, there exists $x\in V(G)$, such that $H(x)\neq H^{\prime}(x)$. Without loss of generality, we may assume that $H(x)=\{1\}$ and $H^{\prime}(x)=\{0,2\}$. By Definition 6 (iii), we have $h_{H}(x)=1$ and $h_{H^{\prime}}(x)=2$, which implies $h_{H}\neq h_{H^{\prime}}$. Next we show $J$ is a surjection. For every $h\in\mathcal{F}$, we may define $H:V(G)\rightarrow\{\{1\},\{0,2\}\}$ such that

[TABLE]

Since $|h^{-1}(1)|$ is even, $|H^{-1}(1)|=|h^{-1}(1)\cap V(G)|$ is even. So we get that $H\in\mathcal{H}$ and $J(H)=h$. Thus $J$ is surjective.

This completes the proof.  $\Box$

Proof of Theorem 3. Let $G$ be a connected cubic graph. By Theorem 5, $G$ is almost 1-tough if and only if $G$ contains an $H$-factor for any $H:V(G)\rightarrow\{\{1\},\{0,2\}\}$ with $|H^{-1}(1)|$ even. Thus by Lemma 10, one can see that $G$ is almost 1-tough if and only if $G_{L}$ contains all $(g,f)$-factors, where $g\equiv 1$ and

[TABLE]

Since it is NP-hard to recognize almost 1-tough graphs by Lemma 8, one can see that it is NP-hard to determine whether a graph contains all $(g,f)$-factors. This completes the proof of Theorem 3.  $\Box$