## 数学代写|组合优化代写Combinatorial optimization代考|Chordal Graphs

In this section, we will consider chordal graphs and characterise those that are contact $B_0$-VPG. First, let us point out the following important observation.
Observation 3. A chordal contact $B_0-V P G$ graph is a block graph.
This follows directly from Lemma 1 and the definition of block graphs.
The following lemma states an important property of minimal chordal non contact $B_0$-VPG graphs that contain neither $K_5$ nor $K_4$-e.

Lemma 2. Let $G$ be a chordal $\left{K_5, K_4\right.$-e $}$-free graph. If $G$ is a minimal non contact $B_0$-VPG graph, then every simplicial vertex of $G$ has degree exactly three.
Proof. Since $G$ is $K_5$-free, every clique in $G$ has size at most four. Therefore, every simplicial vertex has degree at most three. Let $v$ be a simplicial vertex of $G$. Assume first that $v$ has degree one and consider a contact $B_0$-VPG representation of $G-v$ (which exists since $G$ is minimal non contact $B_0-\mathrm{VPG}$ ). Let $w$ be the unique neighbour of $v$ in $G$. Without loss of generality, we may assume that the path $P_w$ lies on some row of the grid. Now clearly, we can add one extra column to the grid between any two consecutive vertices of the grid belonging to $P_w$ and adapt all paths without changing the intersections (if the new column is added bètweeen column $y_i$ and $y_{i+1}$, wè extend all paths containing a grid-êdgee with endpoints in column $y_i$ and $y_{i+1}$ in such a way that they contain the new edges in the same row and between column $y_i$ and $y_{i+2}$ of the new grid, and any other path remains the same). But then we may add a path representing $v$ on this column which only intersects $P_w$ (adding a row to the grid and adapting the paths again, if necessary) and thus, we obtain a contact $B_0$ VPG representation of $G$, a contradiction. So suppose now that $v$ has degree two, and again consider a contact $B_0-\mathrm{VPG}$ representation of $G-v$.

## 数学代写|组合优化代写Combinatorial optimization代考|Algorithm in the Vertex-Arrival Model

We first give an algorithm, $\operatorname{Deg} \operatorname{TEsT}(d, \epsilon)$, which with high probability returns a $(1+\epsilon)$-approximation of $n_d$ using $\mathrm{O}\left(\frac{1}{\epsilon^2} \log ^2 n\right)$ bits of space. In the description of the algorithm, we suppose that we have a random function COIN: $[0,1] \rightarrow{$ false, true $}$ such that $\operatorname{COIN}(p)=$ true with probability $p$ and $\operatorname{COIN}(p)$ $=$ false with probability $1-p$. Furthermore, the outputs of repeated invocations of COIN are independent.

Algorithm DegTest $(d, \epsilon)$ maintains a sample $S$ of at most $c \log n$ vertices. It ensures that all vertices $v \in S$ have degree at most $d$ in the current graph $G_i$ (notice that $\operatorname{deg}{G_i}(v) \leq \operatorname{deg}{G_j}(v)$, for every $j \geq i$ ). Initially, $p=1$, and all vertices of degree at most $d$ are stored in $S$. Whenever $S$ reaches the limiting size of $c \log n$, we downsample $S$ by removing every element of $S$ with probability $\frac{1}{1+\epsilon^{\prime}}$ and update $p \leftarrow p /\left(1+\epsilon^{\prime}\right)$. This guarantees that throughout the algorithm $S$ constitutes a uniform random sample of all vertices of degree at most $d$ in $G_i$.
The algorithm outputs $m \leftarrow c \log (n) / p$ as the estimate for $n_d$, where $p$ is the smallest value of $p$ that occurs during the course of the algorithm. It is updated whenever $S$ reaches the size $c \log n$, since $S$ is large enough at this moment to be used as an accurate predictor for $n_{d, i}$, and hence also for $n_d$.

Lemma 2. Let $0<\epsilon \leq 1$. DEGTEST $(d, \epsilon)$ (Algorithm 1) approximates $n_d$ within a factor $1+\epsilon$ with high probability, i.e.,
$$\frac{n_d}{1+\epsilon} \leq \operatorname{DEGTEST}(d, \epsilon) \leq(1+\epsilon) n_d,$$
and uses $\mathrm{O}\left(\frac{1}{\epsilon^2} \log ^2 n\right)$ bits of space.
For space reasons, we defer the proof of this Lemma to the full version of this paper and only give a brief outline here. We say that the algorithm is in phase $i$ if the current value of $p$ is $p=1 /\left(1+\epsilon^{\prime}\right)^i$.

# 组合优化代考

## 数学代写|组合优化代写Combinatorial optimization代考|Algorithm in the Vertex-Arrival Model

\frac{n_d}{1+\epsilon} \leq \operatorname{DEGTEST}(d, \epsilon) \leq(1+\epsilon) n_d


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