If \(u\) and \(v\) are vertices of a graph, then \(d(u,v)\) denotes the distance from \(u\) to \(v\). Let \(S = \{v_1, v_2, \ldots, v_k\}\) be a set of vertices in a connected graph \(G\). For each \(v \in V(G)\), the \(k\)-vector \(c_S(v)\) is defined by \(c_S(v) = (d(v, v_1), d(v, v_2), \ldots, d(v, v_k))\). A dominating set \(S = \{v_1, v_2, \ldots, v_k\}\) in a connected graph \(G\) is a metric-locating-dominating set, or an MLD-set, if the \(k\)-vectors \(c_S(v)\) for \(v \in V(G)\) are distinct. The metric-location-domination number \(\gamma_M(G)\) of \(G\) is the minimum cardinality of an MLD-set in \(G\). We determine the metric-location-domination number of a tree in terms of its domination number. In particular, we show that \(\gamma(T) = \gamma_M(T)\) if and only if \(T\) contains no vertex that is adjacent to two or more end-vertices. We show that for a tree \(T\) the ratio \(\gamma_L(T)/\gamma_M(T)\) is bounded above by \(2\), where \(\gamma_L(G)\) is the location-domination number defined by Slater (Dominating and reference sets in graphs, J. Math. Phys. Sci. \(22 (1988), 445-455)\). We establish that if \(G\) is a connected graph of order \(n \geq 2\), then \(\gamma_M(G) = n-1\) if and only if \(G = K_{1,n-1}\) or \(G = K_n\). The connected graphs \(G\) of order \(n \geq 4\) for which \(\gamma_M(G) = n-2\) are characterized in terms of seven families of graphs.

The edges of a graph can be either directed or signed (\(2\)-colored) so as to make some of the even-length cycles of the underlying graph into alternating cycles. If a graph has a signing in which every even-length cycle is alternating, then it also has an orientation in which every even-length cycle is alternating, but not conversely. The existence of such an orientation or signing is closely related to the existence of an orientation in which every even-length cycle is a directed cycle.

We deal with the problem of labeling the vertices, edges, and faces of a plane graph in such a way that the label of a face and the labels of the vertices and edges surrounding that face add up to a weight of that face, and the weights of all \(s\)-sided faces constitute an arithmetic progression of difference \(d\). In this paper, we describe various antimagic labelings for the generalized Petersen graph \(P(n, 2)\). The paper concludes with a conjecture.

It was shown by Abrham that the number of pure Skolem sequences of order \(n\), \(n \equiv 0\) or \(1 \pmod{4}\), and the number of extended Skolem sequences of order \(n\), are both bounded below by \(2^{\left\lfloor \frac{n}{3} \right\rfloor}\). These results are extended to give similar lower bounds for the numbers of hooked Skolem sequences, split Skolem sequences, and split-hooked Skolem sequences.

Jin and Liu discovered an elegant formula for the number of rooted spanning forests in the complete bipartite graph \(K_{a_1,a_2}\), with \(b_1\) roots in the first vertex class and \(b_2\) roots in the second vertex class. We give a simple proof of their formula, and a generalization for complete \(m\)-partite graphs, using the multivariate Lagrange inverse.

Using a linear space on \(v\) points with all block sizes \(|B| \equiv 0\) or \(1 \pmod{3}\), Doyen and Wilson construct a Steiner triple system on \(2v+1\) points that embeds a Steiner triple system on \(2|B|+1\) points for each block \(B\). We generalise this result to show that if the linear space on \(v\) points is extendable in a suitable way, there is a Steiner quadruple system on \(2v+2\) points that embeds a Steiner quadruple system on \(2(|B|+1)\) points for each block \(B\).

A graph with a graceful labeling (an \(\alpha\)-labeling) is called a graceful (\(\lambda\)-graceful) graph. In this paper, six methods for constructing bigger graceful graphs from a given graceful graph or a set of given \(\lambda\)-graceful graphs are provided. Two of which generalize Koh and others’ Theorems in [2, 3].

Let \(B_2\) be the bananas surface arising from the torus by contracting two different meridians of the torus to a simple point each. It was proved in [8] that there is not a finite Kuratowski theorem for \(B_2\).

A graph is outer-bananas-surface if it can be embedded in \(B_2\) so that all its vertices lie on the same face. In this paper, we prove that the class of the outer-\(B_2\) graphs is closed under minors. In fact, we give the complete set of \(38\) minor-minimal non-outer-\(B_2\) graphs and we also characterize these graphs by a finite list of forbidden topological minors.

We also extend outer embeddings to other pseudosurfaces. The \(S\) pseudosurfaces treated are spheres joined by points in such a way that each sphere has two singular points. We give an excluded minor characterization of outer-\(S\) graphs and we also give an explicit and finite list of forbidden topological minors for these pseudosurfaces.

We show that several known theorems on graphs and digraphs are equivalent. The list of equivalent theorems include Kotzig’s result on graphs with unique \(1\)-factors, a lemma by Seymour and Giles, theorems on alternating cycles in edge-colored graphs, and a theorem on semicycles in digraphs.

We consider computational problems related to the quoted results; all these problems ask whether a given (di)graph contains a cycle satisfying certain properties which runs through \(p\) prescribed vertices. We show that all considered problems can be solved in polynomial time for \(p < 2\) but are NP-complete for \(p \geq 2\).

We define a new graph operation called “dissolve \(N(v)\) into \(v\)” where \(N(v)\) is the set of vertices adjacent to a vertex \(v\) and characterize odd cycles of length greater than \(5\) in terms of \(p\)-critical graphs using this operation. This enables us to re-phrase the Strong Perfect Graph Conjecture,