Suppose \(G\) and \(G’\) are graphs on the same vertex set \(V\) such that for each \(v \in V\) there is an isomorphism \(\theta_x\) of \(G-v\) to \(G’-v\). We prove in this paper that if there is a vertex \(x \in V\) and an automorphism \(\alpha\) of \(G-x\) such that \(\theta_x\) agrees with \(\alpha\) on all except for at most three vertices of \(V-x\), then \(G\) is isomorphic to \(G’\). As a corollary we prove that if a graph \(G\) has a vertex which is contained in at most three bad pairs, then \(G\) is reconstructible. Here a pair of vertices \(x,y\) of a graph \(G\) is called a bad pair if there exist \(u,v \in V(G)\) such that \(\{u,v\} \neq \{x,y\}\) and \(G-\{x,y\}\) is isomorphic to \(G-\{u,v\}\).

A \(\{0,1\}\)-matrix \(M\) is tree graphic if there exists a tree \(T\) such that the edges of \(T\) are indexed on the rows of \(M\) and the columns are the incidence vectors of the edge sets of paths of \(T\). Analogously, \(M\) is ditree graphic if there exists a ditree \(T\) such that the directed edges of \(T\) are indexed on the rows of \(M\) and the columns are the incidence vectors of the directed-edge sets of dipaths of \(T\). In this paper, a simple proof of an excluded-minor characterization of the class of tree-graphic matrices that are ditree-graphic is given. Then, using the same proof technique, a characterization of a “special” class of tree-graphic matrices (which are contained in the class of consecutive \(1\)’s matrices) is stated and proved.

One of the fundamental results concerning cycles in graphs is due to Ore:
If \(G\) is a graph of order \(n \geq 3\) such that \(d(x) + d(y) \geq n\) for every pair of nonadjacent vertices \(x, y \in V(G)\), then \(G\) is hamiltonian.
We generalize this result using neighborhood unions of \(k\) independent vertices for any fixed integer \(k \geq 1\). That is, for \(A \subseteq V(G)\), let \(N(A) = \cup_{a \in A} N(a),\)
where \(N(a) = \{b : ab \in E(G)\}\) is the neighborhood of \(a\). In particular, we show:
In a \(4(k-1)\)-connected graph \(G\) of order \(n \geq 3\), if \(|N(S)|+|N(T)| \geq n\) for every two disjoint independent vertex sets \(S\) and \(T\) of \(k\) vertices, then \(G\) is hamiltonian.
A similar result for hamiltonian connected graphs is obtained too.

The imbalance of edge \((x,y) = | \deg(x) – \deg(y) |\).The sum of all edge imbalances in a graph is called its irregularity.
We determine the maximum irregularity of various classes of graphs.For example, the irregularity of an arbitrary graph with \(n\) vertices is less than \(\frac{4n^3}{27}\), and this bound is tight.

In this paper we show that simplicial graphs, in which every vertex belongs to exactly one simplex, characterize graphs satisfying equality in some graph invariants concerning independence, clique covering, domination or distance.

The plane in the title is investigated from the combinatorial point of view.Its Baer subplanes are classified and their distribution is studied.Properties of the Fano subplanes are shown.Blocking sets of Rédei type are constructed.
Finally, hyperovals and complete \(14\)-arcs are considered and classified.

A graph \(G\) is \(H\)-decomposable if \(G\) can be decomposed into graphs, each of which is isomorphic to \(H\).
A graph \(G\) without isolated vertices is a least common multiple of two graphs \(G_1\) and \(G_2\) if \(G\) is a graph of minimum size such that \(G\) is both \(G_1\)-decomposable and \(G_2\)-decomposable.
It is shown that two graphs can have an arbitrarily large number of least common multiples.
All graphs \(G\) for which \(G\) and \(P_3\) (and \(G\) and \(2K_2\)) have a unique least common multiple are characterized.
It is also shown that two stars \(K_{1,r}\) and \(K_{1,s}\) have a unique least common multiple if and only if \(r\) and \(s\) are not relatively prime.

A Restricted Resolvable Design \(R_rRP(p, k)\) is a resolvable design on \(p\) points with block sizes \(r\) and \(r+1\) in which each point appears \(\alpha\) times. An \(RRP\) is called uniform if all resolution classes consist of the blocks of the same size.
We show that a uniform \(R_3RP(p,\frac{p}{2} -2)\) exists for all \(p \equiv 12 \mod 24, p \neq 12\) except possibly when \(p = 84\) or \(156\).
We also show that if \(g \equiv 3 \mod 6, g \notin \{3, 21, 39\}\) and \(p = 4g \mod 8g\) then there exists an \(R_3RP(p, \frac{p}{2}-(r+1))\) for all

1. \(r \leq \frac{p-4g}{8g}\) if \(\frac{p}{4g}\) is a prime power congruent to \(1 \mod 6\);
2. \(r \leq \frac{p}{4gq}\) where \(q\) is the smallest proper factor of \(\frac{p}{4g}\) if \(\frac{p}{4g}\) is composite and there exists an \(RT(9, \frac{p}{4gp})\).

The core of a graph was defined by Morgan and Slater [MS80] as a path in the graph minimizing the sum of the distance of all vertices of the graph from the path. A linear algorithm to find the core of a tree has been given in [MS80]. For the general graph the problem can be shown to be NP-hard using a reduction from the Hamiltonian path problem.
A graph with no chordless cycle of length exceeding three is called a chordal graph. Every chordal graph is the intersection graph of a family of subtrees of a tree. The intersection graph of a family of undirected paths of a tree is called a UV graph. The intersection graph of an edge disjoint family of paths of a tree is called a CV graph [AAPX91]. We have characterised that the CV graphs are nothing but block graphs. CV graphs form a proper subclass of UV graphs which in turn form a proper subclass of chordal graphs.
In this paper, we present an \( {O}(ne)\) algorithm to find the core of a CV graph, where \(n\) is the number of vertices and \(e\) is the number of edges.

The worst-case time-complexity of Read’s edge-addition/contraction algorithm for computing the chromatic polynomial of an \(n\)-vertex graph is shown to be \({O}(n^2B(n))\), where \(B(n)\) is the \(n\)th Bell number, which grows faster than \(c^n\) for any \(c\) but slower than \(n!\). The factor \(n^2\) can be reduced to \(n\), and the space-complexity from \({O}(n^3)\) to \({O}(n^2)\), by storing in arrays the information needed to reverse each transformation made on the graph.