C.M. Mynhardt’s research while affiliated with University of Victoria and other places

The independent domination number i ( G ) of a graph G is the minimum cardinality of a maximal independent set of G , also called an i ( G )-set. The i -graph of G , denoted ℐ ( G ), is the graph whose vertices correspond to the i ( G )-sets, and where two i ( G )-sets are adjacent if and only if they differ by two adjacent vertices. Not all graphs are i -graph realizable, that is, given a target graph H , there does not necessarily exist a source graph G such that H = ℐ ( G ). We consider a class of graphs called “theta graphs”: a theta graph is the union of three internally disjoint nontrivial paths with the same two distinct end vertices. We characterize theta graphs that are i -graph realizable, showing that there are only finitely many that are not. We also characterize those line graphs and claw-free graphs that are i -graphs, and show that all 3-connected cubic bipartite planar graphs are i -graphs.

The independent domination number i(G) of a graph G is the minimum cardinality of a maximal independent set of G, also called an i(G)-set. The i-graph of G, denoted $\mathscr{I}(G)$, is the graph whose vertices correspond to the i(G)-sets, and where two i(G)-sets are adjacent if and only if they differ by two adjacent vertices. Although not all graphs are i-graph realizable, that is, given a target graph H, there does not necessarily exist a source graph G such that $H \cong \mathscr{I}(G)$, all graphs have i-graphs. We determine the i-graphs of paths and cycles and, in the case of cycles, discuss the Hamiltonicity of these i-graphs.

The independent domination number i(G) of a graph G is the minimum cardinality of a maximal independent set of G, also called an i(G)-set. The i-graph of G, denoted $\mathcal{I}(G)$, is the graph whose vertices correspond to the i(G)-sets, and where two i(G)-sets are adjacent if and only if they differ by two adjacent vertices. We show that not all graphs are i-graph realizable, that is, given a target graph H, there does not necessarily exist a source graph G such that H is isomorphic to $\mathcal{I}(G)$. Examples of such graphs include $K_{4}-e$ and $K_{2,3}$. We build a series of tools to show that known i-graphs can be used to construct new i-graphs and apply these results to build other classes of i-graphs, such as block graphs, hypercubes, forests, cacti, and unicyclic graphs.

A set D of vertices of a graph G=(V,E) is irredundant if each non-isolated vertex of G[D] has a neighbour in V−D that is not adjacent to any other vertex in D. The upper irredundance number IR(G) is the largest cardinality of an irredundant set of G; an IR(G)-set is an irredundant set of cardinality IR(G). The IR-graph of G has the IR(G)-sets as vertex set, and sets D and D′ are adjacent if and only if D′ can be obtained from D by exchanging a single vertex of D for an adjacent vertex in D′. An IR-tree is an IR-graph that is a tree. We characterize IR-trees of diameter 3 by showing that these graphs are precisely the double stars S(2n,2n), i.e., trees obtained by joining the central vertices of two disjoint stars K1,2n.

A set D of vertices of a graph G=(V,E) is irredundant if each v∈D satisfies (a) v is isolated in the subgraph induced by D, or (b) v is adjacent to a vertex in V−D that is nonadjacent to all other vertices in D. The upper irredundance number IR(G) is the largest cardinality of an irredundant set of G; an IR(G)-set is an irredundant set of cardinality IR(G). The IR-graph of G has the irredundant sets of G of maximum cardinality, that is, the IR(G)-sets, as vertex set, and sets D and D′ are adjacent if and only if D′ is obtained from D by exchanging a single vertex of D for an adjacent vertex in D′. We study the realizability of graphs as IR-graphs and show that all disconnected graphs are IR-graphs, but some connected graphs (e.g. stars K1,n,n≥2, P4,P5,C5,C6,C7) are not. We show that the double star S(2,2) – the tree obtained by joining the two central vertices of two disjoint copies of P3 – is the unique smallest IR-tree with diameter 3 and also a smallest non-complete IR-tree, and the tree obtained by subdividing a single pendant edge of S(2,2) is the unique smallest IR-tree with diameter 4.

We study the relationship between the eternal domination number of a graph and its clique covering number. Using computational methods, we show that the smallest graph having its eternal domination number less than its clique covering number has 10 vertices. This answers a question of Klostermeyer and Mynhardt [Protecting a graph with mobile guards, Appl. Anal. Discrete Math. 10 (2016), no. 1, $1-29$]. We also determine the complete set of 10-vertex and 11-vertex graphs having eternal domination numbers less than their clique covering numbers. In addition, we study the problem on triangle-free graphs, circulant graphs, planar graphs and cubic graphs. Our computations show that all triangle-free graphs and all circulant graphs of order 12 or less have eternal domination numbers equal to their clique covering numbers, and exhibit 13 triangle-free graphs and 2 circulant graphs of order 13 which do not have this property. Using these graphs, we describe a method to generate an infinite family of triangle-free graphs and an infinite family of circulant graphs with eternal domination numbers less than their clique covering numbers. Our computations also show that all planar graphs of order 11 or less, all 3-connected planar graphs of order 13 or less and all cubic graphs of order less than 18 have eternal domination numbers equal to their clique covering numbers. Finally, we show that for any integer $k \geq 2$ there exist infinitely many graphs having domination number and eternal domination number equal to k containing dominating sets which are not eternal dominating sets. This answers another question of Klostermeyer and Mynhardt [Eternal and Secure Domination in Graphs, Topics in domination in graphs, Dev. Math. 64 (2020), $445-478$, Springer, Cham].

A broadcast on a nontrivial connected graph G is a function f from V(G) to the set {0,1,...,diam(G)} such that f(v) is at most the eccentricity of v for all vertices v of G. The weight of f is the sum of the function values over V(G). A vertex u hears f from v if f(v) is positive and u is within distance f(v) from v. A broadcast f is boundary independent if any vertex that hears f from two or more vertices is at distance f(v) from each such vertex v. The maximum weight of a boundary independent broadcast on G is denoted by {\alpha}_{bn}(G). We prove a sharp lower bound on {\alpha}_{bn}(T) for a tree T. Combined with a previously determined upper bound, this gives exact values of {\alpha}_{bn}(T) for some classes of trees T. We also determine {\alpha}_{bn}(T) for trees with exactly two branch vertices and use this result to demonstrate the existence of trees for which {\alpha}_{bn} lies strictly between the lower and upper bounds.

A set D of vertices of a graph G with vertex set V is irredundant if each non-isolated vertex of G[D] has a neighbour in V-D that is not adjacent to any other vertex in D. The upper irredundance number IR(G) is the largest cardinality of an irredundant set of G; an IR(G)-set is an irredundant set of cardinality IR(G). The IR-graph of G has the IR(G)-sets as vertex set, and sets A and B are adjacent if and only if B can be obtained from A by exchanging a single vertex of A for an adjacent vertex in B. An IR-tree is an IR-graph that is a tree. We characterize IR-trees of diameter 3 by showing that these graphs are precisely the double stars S(2n,2n), i.e., trees obtained by joining the central vertices of two disjoint stars K_{1,2n}.

A broadcast on a nontrivial connected graph G with vertex set V is a function f from V to {0,1,...,diam(G)} such that f(v) is at most the eccentricity of v for all v in V. The weight of f is the sum of the function values taken over V. A vertex u hears f from v if f(v) is positive and d(u,v) is at most f(v). A broadcast f is dominating if every vertex of G hears f. The upper broadcast number of G is {\Gamma}_{b}(G), which is the maximum weight of a minimal dominating broadcast on G. A broadcast f is boundary independent if, for any vertex w that hears f from vertices v_{1},...,v_{k}, where k is at least 2, the distance d(w,v_{i}) equals f(v_{i}) for each i. The maximum weight of a boundary independent broadcast is the boundary independence broadcast number {\alpha}_{bn}(G). We compare {\alpha}_{bn} to {\Gamma}_{b}, showing that neither is an upper bound for the other. We show that the differences {\Gamma}_{b}-{\alpha}_{bn} and {\alpha}_{bn}-{\Gamma}_{b} are unbounded, the ratio {\alpha}_{bn}/{\Gamma}_{b} is bounded for all graphs, and {\Gamma}_{b}/{\alpha}_{bn} is bounded for bipartite graphs but unbounded in general.