On star-$k$-PCGs: Exploring class boundaries for small $k$ values
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911779110518784 |
|---|---|
| author | Monti, Angelo Sinaimeri, Blerina |
| author_facet | Monti, Angelo Sinaimeri, Blerina |
| contents | A graph $G=(V,E)$ is a star-$k$-PCG if there exists a weight function $w: V \rightarrow R^+$ and $k$ mutually exclusive intervals $I_1, I_2, \ldots I_k$, such that there is an edge $uv \in E$ if and only if $w(u)+w(v) \in \bigcup_i I_i$. These graphs are related to two important classes of graphs: PCGs and multithreshold graphs. It is known that for any graph $G$ there exists a $k$ such that $G$ is a star-$k$-PCG. Thus, for a given graph $G$ it is interesting to know which is the minimum $k$ such that $G$ is a star-$k$-PCG. We define this minimum $k$ as the star number of the graph, denoted by $γ(G)$. Here we investigate the star number of simple graph classes, such as graphs of small size, caterpillars, cycles and grids. Specifically, we determine the exact value of $γ(G)$ for all the graphs with at most 7 vertices. By doing so we show that the smallest graphs with star number 2 are only 4 and have exactly 5 vertices; the smallest graphs with star number 3 are only 3 and have exactly 7 vertices. Next, we provide a construction showing that the star number of caterpillars is one. Moreover, we show that the star number of cycles and two dimensional grid graphs is 2 and that the star number of $4$-dimensional grids is at least 3. Finally, we conclude with numerous open problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_11860 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On star-$k$-PCGs: Exploring class boundaries for small $k$ values Monti, Angelo Sinaimeri, Blerina Combinatorics Discrete Mathematics 05Cxx G.2.2 A graph $G=(V,E)$ is a star-$k$-PCG if there exists a weight function $w: V \rightarrow R^+$ and $k$ mutually exclusive intervals $I_1, I_2, \ldots I_k$, such that there is an edge $uv \in E$ if and only if $w(u)+w(v) \in \bigcup_i I_i$. These graphs are related to two important classes of graphs: PCGs and multithreshold graphs. It is known that for any graph $G$ there exists a $k$ such that $G$ is a star-$k$-PCG. Thus, for a given graph $G$ it is interesting to know which is the minimum $k$ such that $G$ is a star-$k$-PCG. We define this minimum $k$ as the star number of the graph, denoted by $γ(G)$. Here we investigate the star number of simple graph classes, such as graphs of small size, caterpillars, cycles and grids. Specifically, we determine the exact value of $γ(G)$ for all the graphs with at most 7 vertices. By doing so we show that the smallest graphs with star number 2 are only 4 and have exactly 5 vertices; the smallest graphs with star number 3 are only 3 and have exactly 7 vertices. Next, we provide a construction showing that the star number of caterpillars is one. Moreover, we show that the star number of cycles and two dimensional grid graphs is 2 and that the star number of $4$-dimensional grids is at least 3. Finally, we conclude with numerous open problems. |
| title | On star-$k$-PCGs: Exploring class boundaries for small $k$ values |
| topic | Combinatorics Discrete Mathematics 05Cxx G.2.2 |
| url | https://arxiv.org/abs/2209.11860 |