### 数学代写|组合优化代写Combinatorial optimization代考|Some Families of Split Graphs

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## 数学代写|组合优化代写Combinatorial optimization代考|Some Families of Split Graphs

A graph $G=(C \cup S, E)$ is a split graph if its node set can be partitioned into a clique $C$ and a stable set $S$. Split graphs are closed under taking complements and form the complementary core of chordal graphs since $G$ is a split graph if and only if $G$ and $\bar{G}$ are chordal or if and only if $G$ is $\left(C_{4}, \bar{C}{4}, C{5}\right)$-free [11].
Our aim is to study LTD-sets in some families of split graphs having a regular structure from a polyhedral point of view. Complete Split Graphs. A complete split graph is a split graph where all edges between $C$ and $S$ are present. Complete split graphs can be seen as special case of complete multi-partite graphs studied in Sect. 3. In fact, a complete split graph is a clique if $|S|=1$, a star if $|C|=1$, and a crown if $|C|=2$, see Fig. 3(a), (b). Otherwise, the graph can be seen as a complete multi-partite graph where all parts but one have size 1 , i.e. as $K_{n_{1}, n_{2}, \ldots, n_{p}}$ with $n_{1}=\cdots=n_{p-1}=1$ and $n_{p} \geq 2$ such that $U_{1} \cup \cdots \cup U_{p-1}$ induce the clique $C$ and $U_{p}$ the stable set $S$. Hence, we directly conclude from Lemma 3 and Corollary 5 :

Corollary 6. Let $G=(C \cup S, E)$ be a complete split graph.
(a) If $|S|=1$, then $G$ is a clique,
$$C_{X}(G)=M\left(\mathcal{R}{|C|+1}^{2}\right)$$ and $\gamma{X}(G)=|C|$ for $X \in{O L D, L T D}$.
(b) If $|C|=1$, then $G$ is a star,
$$C_{L T D}(G)=\left(\begin{array}{c|lll} 1 & 0 & \ldots & 0 \ \hline 0 & & \ \vdots & M\left(\mathcal{R}{|S|}^{2}\right) \ 0 & \end{array}\right)$$ and $\gamma{L T D}(G)=|S|$.
(c) Otherwise, we have
$$C_{L T D}(G)=\left(\begin{array}{cc} M\left(\mathcal{R}{|C|}^{2}\right) & 0 \ 0 & M\left(\mathcal{R}{|S|}^{2}\right) \end{array}\right)$$
and $\gamma_{L T D}(G)=|S|+|C|-2$.
Headless Spiders. A headless spider is a split graph with $C=\left{c_{1}, \ldots, c_{k}\right}$ and $S=\left{s_{1}, \ldots, s_{k}\right}$; it is thin (resp. thick) if $s_{i}$ is adjacent to $c_{j}$ if and only if $i=j$ (resp. $i \neq j$ ), see Fig. 3(c), (d) for illustration. Clearly, the complement of a thin spider is a thick spider, and vice-versa. It is easy to see that for $k=2$, the path $P_{4}$ equals the thin and thick headless spider. Moreover, it is easy to check that headless spiders are twin-free.

A thick headless spider with $k=3$ equals the 3 -sun $S_{3}$ and it is easy to see that $\gamma_{O L D}\left(S_{3}\right)=4$ and $\gamma_{L T D}\left(S_{3}\right)=3$ holds. To describe the clutters for $k \geq 4$, we use the following notations. Let $J_{n}$ denote the $n \times n$ matrix having 1-entries only and $I_{n}$ the $n \times n$ identity matrix. Furthermore, let $J_{n-1, n}(i)$ denote a matrix s.t. its $i$-th column has 0 -entries only and removing the $i$-th column results in $J_{n-1}$, and $I_{n-1, n}(j)$ denote a matrix s.t. its $j$-th column has 1 -entries only and removing the $j$-th column results in $I_{n-1}$.

## 数学代写|组合优化代写Combinatorial optimization代考|Concluding Remarks

In this paper, we proposed to study the $O L D$ – and $L T D$-problem from a polyhedral point of view, motivated by promising polyhedral results for the $I D$-problem [2-5]. That way, we were able to provide closed formulas for the LTD-numbers of all kinds of complete $p$-partite graphs (Sect. 3), and for the studied families of split graphs as well as the $O L D$-numbers of thin and thick headless spiders (Sect. 4).

In particular, if we have the same clutter matrix for two different $X$-problems, then we can conclude that every solution of one problem is also a solution for the other problem, and vice versa, such that the two $X$-polyhedra coincide and the two $X$-numbers are equal. This turned out to be the case for

• complete bipartite graphs as $C_{I D}\left(K_{m, n}\right)=C_{L T D}\left(K_{m, n}\right)$ holds by Lemma 2 and results from [2],
• thin headless spiders $G$ as $C_{O L D}(G)=C_{L T D}(G)$ holds by Lemma $5 .$
Furthermore, we were able to provide the complete facet descriptions of
• the LTD-polyhedra for all complete $p$-partite graphs (including complete split graphs) and for thin headless spiders (see Sect. 3 and Lemma 5),
• the $O L D$-polyhedra of cliques, thin and thick headless spiders (see Corollary 5 and Sect. 4).

The complete descriptions of some $X$-polyhedra also provide us with information about the relation between $Q^{}\left(C_{X}(G)\right)$ and its linear relaxation $Q\left(C_{X}(G)\right)$. A matrix $M$ is ideal if $Q^{}(M)=Q(M)$. For any nonideal matrix, we can evaluate how far $M$ is from being ideal by considering the inequalties that have to be added to $Q(M)$ in order to obtain $Q^{}(M)$. With this purpose, in [1], a matrix $M$ is called rank-ideal if only $0 / 1$-valued constraints have to be added to $Q(M)$ to obtain $Q^{}(M)$. From the complete descriptions obtained in Sect. 3 and Sect. 4 , we conclude:

Corollary 9. The LTD-clutters and OLD-clutters of thin headless spiders are ideal for all $k \geq 3$.

Corollary 10. The LTD-clutters of all complete p-partite graphs and the $O L D$ clutters of cliques and thick headless spiders are rank-ideal.

Finally, the LTD-clutters of thick headless spiders have a more complex structure such that also a facet description of the LTD-polyhedra is more involved. However, using polyhedral arguments, is was possible to establish that $k-1$ is a lower bound for the cardinality of any LTD-set. Exhibiting an LTD-set of size $k-1$ thus allowed us to deduce the exact value of the $L T D$-number of thick headless spiders (Theorem 3 ).

This demonstrates how the polyhedral approach can be applied to find $X$ sets of minimum size for special graphs $G$, by determining and analyzing the $X$-clutters $C_{X}(G)$, even in cases where no complete description of $P_{X}(G)$ is known yet.

As future lines of research, we plan to work on a complete description of the LTD-polyhedra of thick headless spiders and to apply similar and more advanced techniques for other graphs in order to obtain either $X$-sets of minimum size or strong lower bounds stemming from linear relaxations of the $X$-polyhedra, enhanced by suitable cutting planes.

## 数学代写|组合优化代写Combinatorial optimization代考|Mourad Ba¨ıou1 and Francisco Barahona2

This paper follows the study of the classical linear formulation for the $p$-median problem started in [1-3]. To avoid repetitions, we refer to [1] for a more detailed introduction on the $p$-median problem.

Let $G=(V, A)$ a directed graph not necessarily connected, where each arc $(u, v) \in A$ has an associated cost $c(u, v)$. Here we make a difference between oriented and directed graphs. In oriented graphs at most one of the the arcs $(u, v)$ or $(v, u)$ exist, while in directed graphs we may have both arcs $(u, v)$ and $(v, u)$. The $p$-median problem $(p \mathrm{MP})$ consists of selecting $p$ nodes, usully called centers, and then assign each nonselected node along an arc to a selected node. The goal is to select $p$ nodes that minimize the sum of the costs yielded by the assignment of the nonselected nodes. If the number of centers is not fixed and in stead we have costs associated with nodes, then we get the well known facility location problem.

If we associate the variables $y$ to the nodes, and the variables $x$ to the arcs, the following is the classical linear relaxation of the $p \mathrm{MP}$. If we remove equality (1), then we get a linear relaxation of the facility location problem.

\begin{aligned} &\sum_{v \in V} y(v)=p, \ &y(u)+\sum_{v:(u, v) \in A} x(u, v)=1 \quad \forall u \in V, \ &x(u, v) \leq y(v) \quad \forall(u, v) \in A, \ &y(v) \geq 0 \quad \forall v \in V, \ &x(u, v) \geq 0 \quad \forall(u, v) \in A . \end{aligned}
Call $p \mathrm{MP}(G)$ the $p$-median polytope, that is the convex hull of integer solutions satisfying (1)-(5).

Now we will introduced a class of valid inequalities based on odd directed cycles. For this we need some additional definitions. A simple cycle $C$ is an ordered sequence $v_{0}, a_{0}, v_{1}, a_{1}, \ldots, a_{t-1}, v_{t}$, where
$-v_{i}, 0 \leq i \leq t-1$, are distinct nodes,

• either $v_{i}$ is the tail of $a_{i}$ and $v_{i+1}$ is the head of $a_{i}$, or $v_{i}$ is the head of $a_{i}$ and $v_{i+1}$ is the tail of $a_{i}$, for $0 \leq i \leq t-1$, and
$-v_{0}=v_{t}$.
Let $V(C)$ and $A(C)$ denote the nodes and the arcs of a simple cycle $C$, respectively. By setting $a_{t}=a_{0}$, we partition the vertices of $C$ into three sets: $\hat{C}, \dot{C}$ and $\vec{C}$. Each node $v$ is incident to two arcs $a^{\prime}$ and $a^{\prime \prime}$ of $C$. If $v$ is the head (resp. tail) of both arcs $a^{\prime}$ and $a^{\prime \prime}$ then $v$ is in $\hat{C}$ (resp. $\dot{C}$ ) and if $v$ is the head of one of them and a tail of the other, then $v$ is in $\tilde{C}$. Notice that $|\hat{C}|=|\dot{C}| . \mathrm{A}$ cycle will be called $g$-odd if $|\tilde{C}|+|\hat{C}|$ is odd, that is the number of nodes that are heads of some arcs in $C$ is odd. Otherwise it will be called $g$-even. A cycle $C$ with $V(C)=\tilde{C}$ is a directed cycle, otherwise it is called a non-directed cycle. Notice that the notion of g-odd (g-even) cycles generalizes the notion of odd (even) directed cycles, that is why we use the letter ” $\mathrm{g}$ “.

## 数学代写|组合优化代写Combinatorial optimization代考|Some Families of Split Graphs

(a) 如果|小号|=1， 然后G是一个集团，
CX(G)=米(R|C|+12)和CX(G)=|C|为了X∈这大号D,大号吨D.
(b) 如果|C|=1， 然后G是一颗星星，
C_{L T D}(G)=\left(\begin{array}{c|lll} 1 & 0 & \ldots & 0 \ \hline 0 & & \ \vdots & M\left(\mathcal{R}{| S|}^{2}\right) \ 0 & \end{数组}\right)C_{L T D}(G)=\left(\begin{array}{c|lll} 1 & 0 & \ldots & 0 \ \hline 0 & & \ \vdots & M\left(\mathcal{R}{| S|}^{2}\right) \ 0 & \end{数组}\right)和C大号吨D(G)=|小号|.
(c) 否则，我们有
C大号吨D(G)=(米(R|C|2)0 0米(R|小号|2))

## 数学代写|组合优化代写Combinatorial optimization代考|Concluding Remarks

• 完全二部图为C一世D(ķ米,n)=C大号吨D(ķ米,n)由引理 2 成立，结果来自 [2]，
• 瘦无头蜘蛛G作为C这大号D(G)=C大号吨D(G)由引理持有5.
此外，我们能够提供完整的方面描述
• 所有完整的 LTD-多面体p- 分图（包括完全分裂图）和瘦无头蜘蛛（见第 3 节和引理 5），
• 这这大号D- 团多面体，薄而厚的无头蜘蛛（见推论 5 和第 4 节）。

## 数学代写|组合优化代写Combinatorial optimization代考|Mourad Ba¨ıou1 and Francisco Barahona2

−在一世,0≤一世≤吨−1, 是不同的节点,

• 任何一个在一世是尾巴一种一世和在一世+1是头一种一世， 或者在一世是头一种一世和在一世+1是尾巴一种一世， 为了0≤一世≤吨−1， 和
−在0=在吨.
让在(C)和一种(C)表示简单循环的节点和弧C， 分别。通过设置一种吨=一种0，我们划分顶点C分为三组：C^,C˙和C→. 每个节点在与两条弧线有关一种′和一种′′的C. 如果在是两个弧的头部（分别是尾部）一种′和一种′′然后在在C^（分别。C˙） 而如果在是其中一个的头和另一个的尾，那么在在C~. 请注意|C^|=|C˙|.一种将调用循环G-如果是奇数|C~|+|C^|是奇数，即是某些弧的头的节点数C很奇怪。否则会被调用G-甚至。一个循环C和在(C)=C~是有向环，否则称为无向环。请注意，g-奇（g-偶）循环的概念概括了奇（偶）有向循环的概念，这就是我们使用字母“G “.

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