«inflated graph»

k-TUPLE TOTAL DOMINATION IN INFLATED GRAPHS

The inflated graph GI of a graph G with n(G) vertices is obtained from G by replacing every vertex of degree d of G by a clique, which is isomorph to the complete graph Kd, and each edge (xi, xj) of G is replaced by an edge (u, v) in such a way that u ∈ Xi, v ∈ Xj , and two diffe

The thickness and chromatic number of r-inflated graphs

A graph has thickness t if the edges can be decomposed into t and no fewer planar layers. We study one aspect of a generalization of Ringel’s famous Earth-Moon problem: what is the largest chromatic number of any thickness-2 graph? In particular, given a graph G we consider the r

Inflatable graph properties and naturalization of property tests through strong canonicality

We consider natural graph property tests, which act entirely independently of the size of the graph being tested (not just having a number of queries independent of the size). We introduce the notion of graph properties being inflatable — closed under taking (balanced) blowups —

More results on r-inflated graphs: Arboricity, thickness, chromatic number and fractional chromatic number

The r-inflation of a graph G is the lexicographic product G with Kr. A graph is said to have thickness t if its edges can be partitioned into t sets, each of which induces a planar graph, and t is smallest possible. In the setting of the r-inflation of planar graphs, we investiga

Inflatable Graph Properties and Natural Property Tests

We consider natural graph property tests, which act entirely independently of the size of the graph being tested (not just having a number of queries independent of the size). We introduce the notion of graph properties being inflatable — closed under taking (balanced) blowups —

Paired-domination in inflated graphs

The inflation GI of a graph G with n(G) vertices and m(G) edges is obtained from G by replacing every vertex of degree d of G by a clique Kd. A set S of vertices in a graph G is a paired dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph

total domination in $k_r$-covered graphs

the inflation $g_{i}$ of a graph $g$ with $n(g)$ vertices and $m(g)$ edges is obtained from $g$ by replacing every vertex of degree $d$ of $g$ by a clique, which is isomorph to the complete graph $k_{d}$, and each edge $(x_{i},x_{j})$ of $g$ is replaced by an edge $(u,v)$ in such

Total domination in $K_r$-covered graphs

The inflation $G_{I}$ of a graph $G$ with $n(G)$ vertices and $m(G)$ edges is obtained from $G$ by replacing every vertex of degree $d$ of $G$ by a clique, which is isomorph to the complete graph $K_{d}$, and each edge $(x_{i},x_{j})$ of $G$ is replaced by an edge $(u,v)$ in such

Inflated graphs with equal independence number and upper irredundance number

K-tuple total domination in inflated graphs