总细分覆盖和量子步移
Total tessellation cover and quantum walk
摘要：
我们提出了图的全交错量子行走模型和全镶嵌覆盖。该模型使用总镶嵌覆盖的概念来描述步行者的运动，该步行者被允许跳到图的顶点和边缘，与先前的模型相比，在先前的模型中，步行者跳到顶点或边缘。我们建立$T_t(G)$上的界限，这是$G$的总镶嵌覆盖中所需的最小镶嵌数量。我们突出这些下界$T_t(G) \geq ω(G)$和$T_t(G)\geq is(G)+1$，其中$ω(G)$是最大团的大小和$is(G)$是最大诱导星形子图的边数。使用这些界限，我们定义了良好的总镶嵌图与$T_t(G)=ω(G)$或$T_t(G)=is(G)+1$。$k$ -total tessellability问题的目的是判定给定图$G$是否具有$T_t(G) \leq k$。我们表明，$k$ -总可镶嵌性是在$\mathcal{P}$良好的总可镶嵌图。我们建立了以下问题的$\mathcal{NP}$ -完整性时，限制到以下类别：（$is(G)+1$）-总可镶嵌图与$ω(G) = 2$ ; $ω(G)$ -总可镶嵌图$G$与$is(G)+1 = 3$ ; $k$ -总可镶嵌图$G$与$\max\{ω(G), is(G)+1\}$远离$k$ ;和$4$ -总可镶嵌图$G$与$ω(G) = is(G)+1 = 4$。因此，我们建立了二部图，线图的三角形的自由图，泛图，平面图，$(2,1)$弦图的硬度结果。
英文摘要：
We propose the total staggered quantum walk model and the total tessellation cover of a graph. This model uses the concept of total tessellation cover to describe the motion of the walker who is allowed to hop both to vertices and edges of the graph, in contrast with previous models in which the walker hops either to vertices or edges. We establish bounds on $T_t(G)$, which is the smallest number of tessellations required in a total tessellation cover of $G$. We highlight two of these lower bounds $T_t(G) \geq ω(G)$ and $T_t(G)\geq is(G)+1$, where $ω(G)$ is the size of a maximum clique and $is(G)$ is the number of edges of a maximum induced star subgraph. Using these bounds, we define the good total tessellable graphs with either $T_t(G)=ω(G)$ or $T_t(G)=is(G)+1$. The $k$-total tessellability problem aims to decide whether a given graph $G$ has $T_t(G) \leq k$. We show that $k$-total tessellability is in $\mathcal{P}$ for good total tessellable graphs. We establish the $\mathcal{NP}$-completeness of the following problems when restricted to the following classes: ($is(G)+1$)-total tessellability for graphs with $ω(G) = 2$; $ω(G)$-total tessellability for graphs $G$ with $is(G)+1 = 3$; $k$-total tessellability for graphs $G$ with $\max\{ω(G), is(G)+1\}$ far from $k$; and $4$-total tessellability for graphs $G$ with $ω(G) = is(G)+1 = 4$. As a consequence, we establish hardness results for bipartite graphs, line graphs of triangle-free graphs, universal graphs, planar graphs, and $(2,1)$-chordal graphs.
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