On star-$k$-PCGs: Exploring class boundaries for small $k$ values

TL;DR

This work introduces and investigates star-$k$-PCGs, a class of graphs where edges are determined by sums of vertex weights falling into union of $k$ disjoint intervals, bridging PCGs and multithreshold graphs. It defines the star number $\gamma(G)$ as the minimal $k$ with a star-$k$-PCG representation and provides a complete characterization for all graphs with at most seven vertices, revealing the smallest $\gamma=2$ graphs have 5 vertices and the smallest $\gamma=3$ graphs have 7 vertices. The paper proves that all caterpillars have $\gamma=1$, cycles have $\gamma=2$, and 2D grids have $\gamma=1$ or $2$ depending on dimensions, while 4D grids satisfy a lower bound $\gamma\ge 3$, with explicit constructions and algorithms supporting these results. It also introduces two algorithms to verify star numbers on small graphs and outlines numerous open problems, including complexity questions for recognizing star-$k$-PCGs for $k\ge 2$.

Abstract

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.

Figures (10)

• Figure 1: (\ref{['fig:image1a']}) An example of a graph $G$ that is a star-$1$-PCG graph. In (\ref{['fig:image1b']}) and (\ref{['fig:image1c']}) the $1$-witness graph and the $1$-witness star for which $G$ is a star-$1$-PCG with $I_1=[5,8]$.
• Figure 2: (\ref{['fig:image1']}) An example of a $3$-FSeq and (\ref{['fig:image2']}) the corresponding $3$-FP sequence. (\ref{['fig:image3']}) Another example of a $3$-FP sequence which does not lead to the identification of a $3$-FSeq. The dashed lines depict edges that are not in $G$, whereas the solid lines illustrate edges that are included in $G$.
• Figure 3: An illustration of Lemma \ref{['lem:consecutive_neighb']} for a graph $G$. The dashed lines depict edges that are not in $G$, whereas the solid lines illustrate edges that are included in $G$. The vertices $\{y_1,y_2,x,y_3,y_4\}$ appear consecutive in $\sigma(G^w)$.
• Figure 4: The list for all non isomorphic graphs with at most $5$ vertices. The graphs $G_{15}, G_{20}, G_{25}, G_{27}$ are star-$2$-PCGs. The rest of the graphs are all star-$1$-PCGs.
• Figure 5: The only three graphs on 7 vertices that are not star-$2$-PCG. (\ref{['fig:g1']}) $\gamma(G_{536})=3$ by setting $w(a)=7, w(b)=1, w(c)=6, w(d)=4, w(e)=5, w(f)=9, w(g)=8$ and $I_1=[4,10], I_2=[14.14], I_3=[17,17]$. (\ref{['fig:g2']}) $\gamma(G_{662})=3$ by setting $w(a)=5, w(b)=3, w(c)=4, w(d)=1, w(e)=7, w(f)=6, w(g)=2$ and $I_1=[4,5], I_2=[7,8], I_3=[12,13]$. (\ref{['fig:g3']}) $\gamma(G_{963})=3$ by setting $w(a)=5, w(b)=4, w(c)=7, w(d)=2, w(e)=1, w(f)=3, w(g)=4$ and $I_1=[5,8], I_2=[10,10], I_3=[12,13]$.

Theorems & Definitions (35)

• Definition 1
• Definition 2
• proof
• Lemma 1
• Lemma 2
• Definition 3
• Lemma 3
• proof
• Definition 4
• Lemma 4