We construct a Hölder continuous function on the unit interval which coincides in uncountably (in fact continuum) many points with every of total variation smaller than 1 passing through origin. conclude that this has impermeable graph—one key concepts introduced paper—and we present further examples functions both permeable and graphs. Moreover, show typical sense Baire category) have The firs...

We present a system for deriving inclusions between graphs from a set of inclusions between graphs taken as hypotheses. The novel features are the extended notion of graph with an explicitly representation of complement, the more involved definition of the system, and its completeness proof due to the embedding of complements. This is an improvement on former work, where complement was introduc...

Tolerance graphs arise from the intersection of intervals with varying tolerances in a way that generalizes both interval graphs and permutation graphs. In this paper we prove that every tolerance graph is perfect by demonstrating that its complement is perfectly orderable. We show that a tolerance graph cannot contain a chordless cycle of length greater than or equal to 5 nor the complement of...

Let G = (V, E) be a simple graph on vertex set V and define a function f : V → {−1, 1}. The function f is a signed dominating function if for every vertex x ∈ V , the closed neighborhood of x contains more vertices with function value 1 than with −1. The signed domination number of G, γs(G), is the minimum weight of a signed dominating function on G. Let G denote the complement of G. In this pa...

A paired-dominating set of a graph G = (V, E) with no isolated vertex is a dominating set of vertices inducing a graph with a perfect matching. The paired-domination number of G, denoted by γpr (G), is the minimum cardinality of a paired-dominating set of G. We consider graphs of order n ≥ 6, minimum degree δ such that G and G do not have an isolated vertex and we prove that – if γpr (G) > 4 an...

Let F(n,R), where R ⊆ [n], be the family of all forests on the vertex set [n] which consists of exactly |R| trees. Similarly, let T (n,R) be the family of all trees on the vertex set [n+ 1] with the property that NT (n+ 1) = R for every T ∈ T (n,R), i.e. the neighborhood of the vertex labeled with n + 1 is exactly the set of vertices in R. It is easy to see that |F(n,R)| = |T (n,R)| by the foll...

A set S of vertices in a graph G is a connected dominating set if every vertex not in S is adjacent to some vertex in S and the subgraph induced by S is connected. The connected domination number γc(G) is the minimum size of such a set. Let δ(G) = min{δ(G), δ(G)}, where G is the complement of G and δ(G) is the minimum vertex degree. We prove that when G and G are both connected, γc(G) + γc(G) ≤...

In this note, we derive the lower bound on the sum for Wiener index of bipartite graph and its bipartite complement, as well as the lower and upper bounds on this sum for the Randić index and Zagreb indices. We also discuss the quality of these bounds.