A connected graph \(G\) is said to be odd path extendable if for any odd path \(P\) of \(G\), the graph \(G – V(P)\) contains a perfect matching. In this paper, we at first time introduce the concept of odd path extendable graphs. Some simple necessary and sufficient conditions for a graph to be odd path extendable are given. In particular, we show that if a graph is odd path extendable, it is hamiltonian.

In this paper, we give one construction for constructing large harmonious graphs from smaller ones. Subsequently, three families of graphs are introduced and some members of them are shown to be or not to be harmonious.

A graph is called set reconstructible if it is determined uniquely (up to isomorphism) by the set of its vertex-deleted subgraphs. We prove that all graphs are set reconstructible if all \(2\)-connected graphs \(G\) with \(diam(G) = 2\) and all \(2\)-connected graphs \(G\) with \(diam(G) = diam(\overline{G}) = 3\) are set reconstructible.

A function \(f: V(G) \to \{-1,0,1\}\) defined on the vertices of a graph \(G\) is a minus total dominating function (MTDF) if the sum of its function values over any open neighborhood is at least one. That is, for every \(v \in V\), \(f(N(v)) \geq 1\), where \(N(v)\) consists of every vertex adjacent to \(v\). The weight of a MTDF is the sum of its function values over all vertices. A MTDF \(f\) is minimal if there does not exist a MTDF \(g: V(G) \to \{-1,0,1\}\), \(f \neq g\), for which \(g(v) \leq f(v)\) for every \(v \in V\). The upper minus total domination number, denoted by \(\Gamma^{-}_{t}(G)\), of \(G\) is the maximum weight of a minimal MTDF on \(G\). A function \(f: V(G) \to \{-1,1\}\) defined on the vertices of a graph \(G\) is a signed total dominating function (STDF) if the sum of its function values over any open neighborhood is at least one. The signed total domination number, denoted by \(\gamma^{s}_{t}(G)\), of \(G\) is the minimum weight of a STDF on \(G\). In this paper, we establish an upper bound on \(\Gamma^{-}_{t}(G)\) of the 5-regular graph and characterize the extremal graphs attaining the upper bound. Also, we exhibit an infinite family of cubic graphs in which the difference \(\Gamma^{-}_t(G) – \gamma^{s}_t(G)\) can be made arbitrarily large.

Let \(G\) be a graph with vertex set \(V(G)\). An edge coloring \(C\) of \(G\) is called an edge-cover coloring, if for each color, the edges assigned with it form an edge cover of \(G\). The maximum positive integer \(k\) such that \(G\) has a \(k\)-edge-cover coloring is called the edge cover chromatic index of \(G\) and is denoted by \(\chi’_c(G)\). It is well known that \(\min\{d(v) – \mu(v) : v \in V(G)\} \leq \chi’_c(G) \leq \delta(G)\), where \(\mu(v)\) is the multiplicity of \(v\) and \(\delta(G)\) is the minimum degree of \(G\). If \(\chi’_c(G) = \delta(G)\), \(G\) is called a graph of CI class, otherwise \(G\) is called a graph of CII class. In this paper, we give a new sufficient condition for a nearly bipartite graph to be of CI class.

Though the well-known Vizing’s conjecture is not true for directed graphs in general, we show that it is true when the digraph and its reversal contain an efficient dominating set. In this paper, we investigate the existence of such sets in directed tori and infinite grids. We give a complete characterization of efficient dominating sets in the \(3\)-dimensional case and show the nonexistence of efficient \(d\)-dominating sets in directed tori for any \(d > 1\) and any dimension \(n > 1\).

For every two vertices \(u\) and \(v\) in a graph \(G\), a \(u-v\) geodesic is a shortest path between \(u\) and \(v\). Let \(I(u,v)\) denote the set of all vertices lying on a \(u-v\) geodesic. For a vertex subset \(S\), let \(I_G(S)\) denote the union of all \(I_G(u,v)\) for \(u,v \in S\). The geodetic number \(g(G)\) of a graph \(G\) is the minimum cardinality of a set \(S\) with \(I_G(S) = V(G)\). For a digraph \(D\), there is analogous terminology for the geodetic number \(g(D)\). The geodetic spectrum of a graph \(G\), denoted by \(S(G)\), is the set of geodetic numbers over all orientations of graph \(G\). The lower geodetic number is \(g^-(G) = \min S(G)\) and the upper geodetic number is \(g^+(G) = \max S(G)\). The main purpose of this paper is to investigate lower and upper geodetic numbers of graphs. Our main results in this paper are:

1. For every spanning tree \(T\) of a connected graph \(G\), \(g^-(G) \leq \ell(T)\), where \(\ell(T)\) is the number of leaves of \(T\).
2. The conjecture \(g^+(G) \geq g(G)\) is true for chordal graphs, triangle-free graphs and \(4\)-colorable graphs.