A graph and its complement with specified properties. IV. Counting self-complementary blocks (original) (raw)

A graph and its complement with specified properties V: The self‐complement index

Mathematika, 1980

The self-complement index s(G) of a graph G is the maximum order of an induced subgraph of G whose complement is also induced in G. This new graphical invariant provides a measure of how close a given graph is to being selfcomplementary. We establish the existence of graphs G of order p having s(G) = n for all positive integers n < p. We determine s(G) for several families of graphs and find in particular that when G is a tree, s{G) = 4 unless G is a star for which s(G) = 2. §1. The self-complement index and the induced number. Our purpose is to propose invariants which, in some sense, measure the degree to which a graph is selfcomplementary. To this end we define two related invariants which satisfy this requirement. We then show that the two are equivalent. Throughout we use the notation and terminology of [2]. In particular, all graphs are finite, without loops or multiple lines. The order of a graph G is the number p of points in it. And if X is a set of points in a graph G then we use <.Y> to denote the subgraph of G induced by X. The self-complement index of a graph G, denoted s(G), is defined as the order of the largest induced subgraph H of G, such that H is also induced in G. For a graph G of order p it is clear that 1 < s(G) < p as we do not include the null graph in the family of graphs; see Figure 1 in [3]. Now a related invariant of a graph G is defined. The induced number m(G) is the minimum order of a graph which contains both G and G as induced subgraphs. The first result indicates that s(G) and m(G) are essentially identical. We then proceed to show that s(G) partitions the graphs of order p into p classes which are nonempty except when s(G) = p and p = 2 or 3 (mod 4). The number s(G) is then derived for several important families of graphs. §2. The equivalence of the two invariants. THEOREM 1. If G is a graph of order p with self-complement index s(G) and induced number m{G), then m{G) = 2p-s(G). Proof. Let s = s(G), m = w(G), and let H of order s be a largest induced subgraph of G whose complement is also induced in G. To prove the upper bound, we construct a graph F of order 2p-s in which both G and G are induced subgraphs. Consider disjoint copies of G and G. Let U c V(G)

On almost self-complementary graphs

Discrete Mathematics, 2006

A graph is called almost self-complementary if it is isomorphic to one of its almost complements X c − I, where X c denotes the complement of X and I a perfect matching (1-factor) in X c. Almost self-complementary circulant graphs were first studied by Dobson andŠajna in 2004. In this paper we investigate some of the properties and constructions of general almost self-complementary graphs. In particular, we give necessary and sufficient conditions on the order of an almost self-complementary regular graph, and construct infinite families of almost self-complementary regular graphs, almost selfcomplementary vertex-transitive graphs, and non-cyclically almost self-complementary circulant graphs.

More on almost self-complementary graphs

Discrete Mathematics, 2009

A graph X is called almost self-complementary if it is isomorphic to one of its almost complements X c − I, where X c denotes the complement of X and I a perfect matching (1-factor) in X c. If I is a perfect matching in X c and ϕ : X → X c − I is an isomorphism, then the graph X is said to be fairly almost self-complementary if ϕ preserves I setwise, and unfairly almost self-complementary if it does not. In this paper we construct connected graphs of all possible orders that are fairly and unfairly almost self-complementary, fairly but not unfairly almost self-complementary, and unfairly but not fairly almost self-complementary, respectively, as well as regular graphs of all possible orders that are fairly and unfairly almost self-complementary. Two perfect matchings I and J in X c are said to be X-non-isomorphic if no isomorphism from X + I to X + J induces an automorphism of X. We give a constructive proof to show that there exists a graph X that is almost self-complementary with respect to two X-nonisomorphic perfect matchings for every even order greater than or equal to four.

The degree sequences of self-complementary graphs

Journal of Combinatorial Theory, Series B, 1976

Known necessary conditions for realization of a sequence of integers as the degrees of a self-complementary graph are shown to he sufficient. An algorithm for constructing a realization of such a sequence as degrees of such a graph is illustrated by an example.

Self-complementary graphs and generalisations: A comprehensive reference manual

University of Malta, 1999

A graph which is isomorphic to its complement is said to be a self-complementary graph, or sc-graph for short. These graphs have a high degree of structure, and yet they are far from trivial. Suffice to say that the problem of recognising self-complementary graphs, and the problem of checking two sc-graphs for isomorphism, are both equivalent to the graph isomorphism problem.

On the Generalized Complement of Some Graphs

Asia Pacific Journal of Mathematics, 2021

A. In this paper we study the generalized complement of the graph G m,n = (V, E) for some values of m, n. We study the generalized complement of G m,n graphs with respect to the equal degree partition. The 2−complement of G m,n graphs are also determined for m = 2, n is even or odd. In particular, for some values of m, n ∈ N, we studied the complement of G m,n graphs with respect to the equal degree partition and the 2−complement of G m,n graphs. We determine the partitions P k , k ∈ N of the vertex set V such that the generalized complement of G m,n graph is a path graph and a comb graph.

A graph and its complement with specified properties I: connectivity

International Journal of Mathematics and Mathematical Sciences, 1979

Dedicated to Karl Menger ABSTRACT. We investigate the conditions under which both a graph G and its complement G possess a specified property. In particular, we characterize all graphs G for which G and G both (a) have connectivity one, (b) have line-connectivity one, (c) are 2-connected, (d) are forests, (e) are bipartite, (f) are outerplanar and (g) are eulerlan. The proofs are elementary but amusing. KEF WORDS AND PHRASES. Graphs, Complement. AMS (MOS) SUBJECT CLASSIFICATION (1970) CODES. 05C99.

On self-complementation

Journal of Graph Theory, 1985

We prove that, with very few exceptions, every graph of order n, n = 0, 1 (mod 4) and size a t most n-1, is contained in a self-complementary graph of order n. We study a similar problem for digraphs. Throughout the paper, G and D will denote a finite graph and a finite digraph, respectively, without loops or multiple edges, with vertex-sets V (G) and V (D) , and edge-sets E(G) and E(D); define r (G) = IE(G)I, e(D) = (E(D)I. An edge of G joining x and y is denoted by xy, an edge of D from i to t by (z , t) , and a symmetric edge of D joining u and u by uu. Cj denotes a cycle of G of length i z= 3. G U H will refer to two vertex disjoint graphs G and H , and mG to m disjoint copies of G. If A C V(G), then G-A is the subgraph of G induced byV(G)-A. G is said to be a self-complementary graph (or S.C. graph) if it is isomorphic to its complement c, then, there exists a permutation (T of V(G), called S.C. permutation of G, such that xy is an edge of G if and only if a (x) u (y) is an edge of (for simplicity, we use the notation (~(x y) = u(x)cr(y)). We say that G is contained in a graph G' if there exists a subgraph in G' isomorphic to G. The same is applicable to D. It is known [6,7,8] that if G is an S.C. graph of order n , then n = 0, 1 (mod 4), and its S.C. permutation has all its cycles of lengths being multiples of 4 (except one of length one if n is odd), and lengths of cycles of an S.C. permutation of an S. C. digraph D are even (except one of length one if the order of D is odd). Furthermore, if G is an S.C.

Certain classes of complementary equienergetic graphs

Transactions on Combinatorics, 2021

The energy of a graph is the sum of the absolute values of the eigenvalues of a graph. Two graphs are said to be equienergetic if they have same energy. A graph is said to be complementary equienergetic if it is equienergetic with its complement. Recently several complementary equienergetic graphs have been identified. In this paper, we characterize the cycle, path, complete bipartite regular graph and iterated line graph of regular graph, which are complementary equienergetic.

On the Complementary Equienergetic Graphs

arXiv: Combinatorics, 2019

Energy of a simple graph GGG, denoted by mathcalE(G)\mathcal{E}(G)mathcalE(G), is the sum of the absolute values of the eigenvalues of GGG. Two graphs with the same order and energy are called equienergetic graphs. A graph GGG with the property GcongoverlineGG\cong \overline{G}GcongoverlineG is called self-complementary graph, where overlineG\overline{G}overlineG denotes the complement of GGG. Two non-self-complementary equienergetic graphs G1G_1G1 and G2G_2G2 satisfying the property G1congoverlineG2G_1\cong \overline{G_2}G_1congoverlineG_2 are called complementary equienergetic graphs. Recently, Ramane et al. [Graphs equienergetic with their complements, MATCH Commun. Math. Comput. Chem. 82 (2019) 471-480] initiated the study of the complementary equienergetic regular graphs and they asked to study the complementary equienergetic non-regular graphs. In this paper, by developing some computer codes and by making use of some software like Nauty, Maple and GraphTea, all the complementary equienergetic graphs with at most 10 vertices as well as all the members of the graph class $\...